[ { "question_id": 8, "problem_name": "stochastic_processes_martingale_1", "informal_statement": "Consider biased gambler’s ruin: at each step, the gambler gains one dollar\n with probability p and losses one dollar with probability (1 − p).\n Let X_n be the money in purse at time n.\n\n Show that if p = 1/2, then (X_n) is a martingale.", "lean_statement": "theorem q8_gambler_martingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hadap : Adapted ℱ X) (hint : ∀ n, Integrable (X n) μ) (p : ℝ) (hp : p = 1 / 2)\n (hstep : IsConstDrift μ ℱ X (2 * p - 1)) :\n Martingale X ℱ μ := by sorry", "true_to_source": "literal", "proof": "theorem q8_gambler_martingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hadap : Adapted ℱ X) (hint : ∀ n, Integrable (X n) μ) (p : ℝ) (hp : p = 1 / 2)\n (hstep : IsConstDrift μ ℱ X (2 * p - 1)) :\n Martingale X ℱ μ := by\n apply martingale_of_condExp_sub_eq_zero_nat hadap.stronglyAdapted hint\n intro i\n refine (hstep i).trans ?_\n refine ae_of_all μ (fun ω => ?_)\n simp only [Pi.zero_apply]\n rw [hp]; norm_num", "time_seconds": 40, "topic": "Martingales & stopping" }, { "question_id": 10, "problem_name": "stochastic_processes_birth_death_reversible", "informal_statement": "Consider a birth-death Markov chain on a countable state space that possesses a stationary distribution.\n\nShow that the chain is reversible.", "lean_statement": "theorem q10_birth_death_reversible\n (P : ℕ → ℕ → ℝ) (hP0 : ∀ i j, 0 ≤ P i j) (hP1 : ∀ i, ∑' j, P i j = 1)\n (hbd : ∀ i j, 1 < max i j - min i j → P i j = 0)\n (π : ℕ → ℝ) (hπ0 : ∀ i, 0 ≤ π i) (hπsum : ∑' i, π i = 1)\n (hstat : ∀ j, ∑' i, π i * P i j = π j) :\n ∀ i j, π i * P i j = π j * P j i := by\n sorry", "true_to_source": "abstract", "proof": "theorem q10_birth_death_reversible\n (P : ℕ → ℕ → ℝ) (hP0 : ∀ i j, 0 ≤ P i j) (hP1 : ∀ i, ∑' j, P i j = 1)\n (hbd : ∀ i j, 1 < max i j - min i j → P i j = 0)\n (π : ℕ → ℝ) (hπ0 : ∀ i, 0 ≤ π i) (hπsum : ∑' i, π i = 1)\n (hstat : ∀ j, ∑' i, π i * P i j = π j) :\n ∀ i j, π i * P i j = π j * P j i := by\n -- P vanishes for non-adjacent states\n have hbd' : ∀ a b : ℕ, a + 1 < b ∨ b + 1 < a → P a b = 0 := by\n intro a b h\n apply hbd\n rcases le_total a b with hle | hle\n · rw [max_eq_right hle, min_eq_left hle]; omega\n · rw [max_eq_left hle, min_eq_right hle]; omega\n -- tsum over a two-point support\n have two_sum : ∀ (f : ℕ → ℝ) (a b : ℕ), a < b →\n (∀ k, k ≠ a → k ≠ b → f k = 0) → ∑' k, f k = f a + f b := by\n intro f a b hab hf\n have hsupp : ∀ k ∉ ({a, b} : Finset ℕ), f k = 0 := by\n intro k hk\n simp only [Finset.mem_insert, Finset.mem_singleton] at hk\n push_neg at hk\n exact hf k hk.1 hk.2\n rw [tsum_eq_sum hsupp,\n Finset.sum_insert (by simp only [Finset.mem_singleton]; omega),\n Finset.sum_singleton]\n -- tsum over a three-point support\n have three_sum : ∀ (f : ℕ → ℝ) (a b c : ℕ), a < b → b < c →\n (∀ k, k ≠ a → k ≠ b → k ≠ c → f k = 0) → ∑' k, f k = f a + f b + f c := by\n intro f a b c hab hbc hf\n have hsupp : ∀ k ∉ ({a, b, c} : Finset ℕ), f k = 0 := by\n intro k hk\n simp only [Finset.mem_insert, Finset.mem_singleton] at hk\n push_neg at hk\n exact hf k hk.1 hk.2.1 hk.2.2\n rw [tsum_eq_sum hsupp,\n Finset.sum_insert (by simp only [Finset.mem_insert, Finset.mem_singleton]; omega),\n Finset.sum_insert (by simp only [Finset.mem_singleton]; omega),\n Finset.sum_singleton]\n ring\n -- detailed balance for adjacent states, by induction\n have hadj : ∀ i, π i * P i (i + 1) = π (i + 1) * P (i + 1) i := by\n intro i\n induction i with\n | zero =>\n show π 0 * P 0 1 = π 1 * P 1 0\n have hsupp0 : ∀ k, k ≠ 0 → k ≠ 1 → π k * P k 0 = 0 := by\n intro k h1 h2; rw [hbd' k 0 (by omega)]; ring\n have hsuppR0 : ∀ j, j ≠ 0 → j ≠ 1 → P 0 j = 0 := by\n intro j h1 h2; exact hbd' 0 j (by omega)\n have stat0 : π 0 * P 0 0 + π 1 * P 1 0 = π 0 := by\n have h : (∑' k, π k * P k 0) = π 0 * P 0 0 + π 1 * P 1 0 :=\n two_sum (fun k => π k * P k 0) 0 1 (by omega) hsupp0\n rw [← h]; exact hstat 0\n have row0 : P 0 0 + P 0 1 = 1 := by\n have h : (∑' j, P 0 j) = P 0 0 + P 0 1 :=\n two_sum (fun j => P 0 j) 0 1 (by omega) hsuppR0\n rw [← h]; exact hP1 0\n have keyB : π 0 * P 0 0 + π 0 * P 0 1 = π 0 := by\n have expand : π 0 * (P 0 0 + P 0 1) = π 0 * P 0 0 + π 0 * P 0 1 := by ring\n rw [← expand, row0, mul_one]\n linarith [stat0, keyB]\n | succ i ih =>\n show π (i + 1) * P (i + 1) (i + 2) = π (i + 2) * P (i + 2) (i + 1)\n have hsuppS : ∀ k, k ≠ i → k ≠ i + 1 → k ≠ i + 2 → π k * P k (i + 1) = 0 := by\n intro k h1 h2 h3; rw [hbd' k (i + 1) (by omega)]; ring\n have hsuppRS : ∀ j, j ≠ i → j ≠ i + 1 → j ≠ i + 2 → P (i + 1) j = 0 := by\n intro j h1 h2 h3; exact hbd' (i + 1) j (by omega)\n have statS : π i * P i (i + 1) + π (i + 1) * P (i + 1) (i + 1)\n + π (i + 2) * P (i + 2) (i + 1) = π (i + 1) := by\n have h : (∑' k, π k * P k (i + 1))\n = π i * P i (i + 1) + π (i + 1) * P (i + 1) (i + 1)\n + π (i + 2) * P (i + 2) (i + 1) :=\n three_sum (fun k => π k * P k (i + 1)) i (i + 1) (i + 2) (by omega) (by omega) hsuppS\n rw [← h]; exact hstat (i + 1)\n have rowS : P (i + 1) i + P (i + 1) (i + 1) + P (i + 1) (i + 2) = 1 := by\n have h : (∑' j, P (i + 1) j)\n = P (i + 1) i + P (i + 1) (i + 1) + P (i + 1) (i + 2) :=\n three_sum (fun j => P (i + 1) j) i (i + 1) (i + 2) (by omega) (by omega) hsuppRS\n rw [← h]; exact hP1 (i + 1)\n have keyS : π (i + 1) * P (i + 1) i + π (i + 1) * P (i + 1) (i + 1)\n + π (i + 1) * P (i + 1) (i + 2) = π (i + 1) := by\n have expand : π (i + 1) * (P (i + 1) i + P (i + 1) (i + 1) + P (i + 1) (i + 2))\n = π (i + 1) * P (i + 1) i + π (i + 1) * P (i + 1) (i + 1)\n + π (i + 1) * P (i + 1) (i + 2) := by ring\n rw [← expand, rowS, mul_one]\n linarith [statS, keyS, ih]\n -- assemble for all i, j\n intro i j\n rcases lt_trichotomy i j with hlt | heq | hgt\n · rcases Nat.lt_or_ge (i + 1) j with h2 | h2\n · rw [hbd' i j (Or.inl h2), hbd' j i (Or.inr h2)]; ring\n · have hj : j = i + 1 := by omega\n subst hj; exact hadj i\n · rw [heq]\n · rcases Nat.lt_or_ge (j + 1) i with h2 | h2\n · rw [hbd' i j (Or.inr h2), hbd' j i (Or.inl h2)]; ring\n · have hi : i = j + 1 := by omega\n subst hi; exact (hadj j).symm", "time_seconds": 149, "topic": "Markov chains (finite & countable)" }, { "question_id": 11, "problem_name": "stochastic_processes_birth_death_process_stationary", "informal_statement": "Consider an irreducible birth-death continuous-time Markov chain with birth rate λ_i in state i and death rate μ_{i+1} from state i+1, and let ρ_i = λ_i/μ_{i+1}.\n\nShow that the steady-state distribution, when it exists, satisfies p_i = p_0 ∏_{j exact hbal0\n | succ n ih => linear_combination hbal n + ih\n have hstep : ∀ n, p (n + 1) = p n * ρ n := by\n intro n\n have hm : mu (n + 1) ≠ 0 := (hmu n).ne'\n rw [hρ n, ← mul_div_assoc]\n exact (eq_div_iff hm).mpr (key n).symm\n refine ⟨key, ?_⟩\n intro i\n induction i with\n | zero => simp\n | succ n ih => rw [Finset.prod_range_succ, ← mul_assoc, ← ih, hstep n]", "time_seconds": 130, "topic": "Continuous-time Markov & queues" }, { "question_id": 19, "problem_name": "stochastic_processes_martingale_stopping_time_1", "informal_statement": "Let (X_n)_{n ≥ 0} be a martingale, and let S ≤ T be bounded stopping times.\n\n Show that 𝔼[X_T] = 𝔼[X_S].", "lean_statement": "theorem q19_optional_stopping_eq (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Martingale X ℱ μ) (S T : Ω → ℕ)\n (hS : IsStoppingTime ℱ (natStop S))\n (hT : IsStoppingTime ℱ (natStop T))\n (hST : ∀ ω, S ω ≤ T ω) (N : ℕ) (hTbdd : ∀ ω, T ω ≤ N) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ\n = ∫ ω, stoppedValue X (natStop S) ω ∂μ := by sorry", "true_to_source": "literal", "proof": "theorem q19_optional_stopping_eq (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Martingale X ℱ μ) (S T : Ω → ℕ)\n (hS : IsStoppingTime ℱ (natStop S))\n (hT : IsStoppingTime ℱ (natStop T))\n (hST : ∀ ω, S ω ≤ T ω) (N : ℕ) (hTbdd : ∀ ω, T ω ≤ N) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ\n = ∫ ω, stoppedValue X (natStop S) ω ∂μ := by\n have hle : natStop S ≤ natStop T := by\n intro ω\n unfold natStop\n exact WithTop.coe_le_coe.mpr (hST ω)\n have hbdd : ∀ ω, natStop T ω ≤ (N : WithTop ℕ) := by\n intro ω\n unfold natStop\n exact WithTop.coe_le_coe.mpr (hTbdd ω)\n have h1 := hX.submartingale.expected_stoppedValue_mono hS hT hle hbdd\n have h2 := hX.neg.submartingale.expected_stoppedValue_mono hS hT hle hbdd\n have hnegS : stoppedValue (-X) (natStop S) = fun ω => -(stoppedValue X (natStop S) ω) := rfl\n have hnegT : stoppedValue (-X) (natStop T) = fun ω => -(stoppedValue X (natStop T) ω) := rfl\n rw [hnegS, hnegT] at h2\n simp only [integral_neg] at h2\n linarith [h1, h2]", "time_seconds": 300, "topic": "Martingales & stopping" }, { "question_id": 22, "problem_name": "stochastic_processes_martingale_2", "informal_statement": "Let (ξ_i)_{i ≥ 1} be i.i.d with 𝔼[ξ1] = 0.\n\n Show that X_n = sum_1^n ξ_i is a martingale.", "lean_statement": "theorem q22_martingale_sum (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (ξ : ℕ → Ω → ℝ) (hint : ∀ n, Integrable (ξ n) μ)\n (hmeas : ∀ n, StronglyMeasurable[ℱ (n + 1)] (ξ n))\n (hmean : ∀ n, μ[ξ n | ℱ n] =ᵐ[μ] 0) :\n Martingale (fun n ω => ∑ i ∈ Finset.range n, ξ i ω) ℱ μ := by sorry", "true_to_source": "literal", "proof": "theorem q22_martingale_sum (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (ξ : ℕ → Ω → ℝ) (hint : ∀ n, Integrable (ξ n) μ)\n (hmeas : ∀ n, StronglyMeasurable[ℱ (n + 1)] (ξ n))\n (hmean : ∀ n, μ[ξ n | ℱ n] =ᵐ[μ] 0) :\n Martingale (fun n ω => ∑ i ∈ Finset.range n, ξ i ω) ℱ μ := by\n apply martingale_of_condExp_sub_eq_zero_nat\n · intro n\n have h : ((fun n ω => ∑ i ∈ Finset.range n, ξ i ω) n)\n = (∑ i ∈ Finset.range n, ξ i : Ω → ℝ) :=\n funext fun ω => (Finset.sum_apply ω _ ξ).symm\n rw [h]\n apply Finset.stronglyMeasurable_sum\n intro i hi\n exact (hmeas i).mono (ℱ.mono (Nat.succ_le_of_lt (Finset.mem_range.mp hi)))\n · intro n\n exact integrable_finsetSum _ (fun i _ => hint i)\n · intro i\n refine (condExp_congr_ae ?_).trans (hmean i)\n filter_upwards with ω\n simp [Pi.sub_apply, Finset.sum_range_succ]", "time_seconds": 259, "topic": "Martingales & stopping" }, { "question_id": 24, "problem_name": "stochastic_processes_ctmc_embedded_vs_process_probabilities", "informal_statement": "Let (X(t)) be an irreducible positive-recurrent continuous-time Markov chain with embedded stationary distribution (π_j) and exit rates (ν_j), and process probabilities (p_j).\n\nShow that p_j = π_j when all exit rates ν_j are equal, and give an example where the two distributions differ.", "lean_statement": "theorem q24_ctmc_embedded_vs_process {S : Type*} [Countable S]\n (P : S → S → ℝ) (hP0 : ∀ i j, 0 ≤ P i j) (hPstoch : ∀ i, ∑' j, P i j = 1)\n (hPloop : ∀ i, P i i = 0)\n (π : S → ℝ) (hπpos : ∀ i, 0 < π i) (hπsum : ∑' i, π i = 1)\n (hπstat : ∀ j, ∑' i, π i * P i j = π j)\n (ν : S → ℝ) (hν : ∀ i, 0 < ν i)\n (ν0 : ℝ) (hνeq : ∀ i, ν i = ν0)\n (hsum : Summable (fun k => π k / ν k))\n (p : S → ℝ) (hp : ∀ j, p j = (π j / ν j) / (∑' k, π k / ν k)) :\n ∀ j, p j = π j := by\n sorry", "true_to_source": "abstract", "proof": "theorem q24_ctmc_embedded_vs_process {S : Type*} [Countable S]\n (P : S → S → ℝ) (hP0 : ∀ i j, 0 ≤ P i j) (hPstoch : ∀ i, ∑' j, P i j = 1)\n (hPloop : ∀ i, P i i = 0)\n (π : S → ℝ) (hπpos : ∀ i, 0 < π i) (hπsum : ∑' i, π i = 1)\n (hπstat : ∀ j, ∑' i, π i * P i j = π j)\n (ν : S → ℝ) (hν : ∀ i, 0 < ν i)\n (ν0 : ℝ) (hνeq : ∀ i, ν i = ν0)\n (hsum : Summable (fun k => π k / ν k))\n (p : S → ℝ) (hp : ∀ j, p j = (π j / ν j) / (∑' k, π k / ν k)) :\n ∀ j, p j = π j := by\n intro j\n have hν0 : 0 < ν0 := by rw [← hνeq j]; exact hν j\n have hstep : ∀ k, π k / ν k = π k / ν0 := fun k => by rw [hνeq k]\n have hsumval : (∑' k, π k / ν k) = 1 / ν0 := by\n simp_rw [hstep]\n rw [tsum_div_const, hπsum]\n rw [hp j, hstep j, hsumval]\n field_simp", "time_seconds": 110, "topic": "Continuous-time Markov & queues" }, { "question_id": 29, "problem_name": "stochastic_processes_round_robin_steady_state", "informal_statement": "Consider the round-robin service system modeled as a Markov chain with arrival probability λ per quantum and geometric service requirement with per-quantum completion probability, and let ρ be the ratio of arrival rate to service rate with ρ < 1.\n\nShow that the steady-state distribution of the number in the system is geometric with parameter ρ.", "lean_statement": "theorem q29_round_robin_steady_state\n (lam mu ρ : ℝ) (hmu : 0 < mu) (hρdef : ρ = lam / mu) (hρ0 : 0 ≤ ρ) (hρ1 : ρ < 1)\n (p q : ℕ → ℝ) (hp : ∀ i, 0 < p i) (hq : ∀ i, 0 < q (i + 1))\n (hratio : ∀ i, p i / q (i + 1) = ρ) :\n ∀ π : ℕ → ℝ, (∀ i, 0 ≤ π i) → (∑' i, π i = 1) →\n (∀ i, π i * p i = π (i + 1) * q (i + 1)) →\n ∀ i, π i = (1 - ρ) * ρ ^ i := by\n sorry", "true_to_source": "abstract", "proof": "theorem q29_round_robin_steady_state\n (lam mu ρ : ℝ) (hmu : 0 < mu) (hρdef : ρ = lam / mu) (hρ0 : 0 ≤ ρ) (hρ1 : ρ < 1)\n (p q : ℕ → ℝ) (hp : ∀ i, 0 < p i) (hq : ∀ i, 0 < q (i + 1))\n (hratio : ∀ i, p i / q (i + 1) = ρ) :\n ∀ π : ℕ → ℝ, (∀ i, 0 ≤ π i) → (∑' i, π i = 1) →\n (∀ i, π i * p i = π (i + 1) * q (i + 1)) →\n ∀ i, π i = (1 - ρ) * ρ ^ i := by\n intro π hπ0 hπsum hdbal\n have hstep : ∀ i, π (i + 1) = π i * ρ := by\n intro i\n have hdb := hdbal i\n have hr := hratio i\n rw [← hr, ← mul_div_assoc, eq_div_iff (ne_of_gt (hq i))]\n linarith [hdb]\n have hpow : ∀ n, π n = π 0 * ρ ^ n := by\n intro n\n induction n with\n | zero => simp\n | succ m ih => rw [hstep m, ih]; ring\n have hgeo : ∑' i : ℕ, ρ ^ i = (1 - ρ)⁻¹ := tsum_geometric_of_lt_one hρ0 hρ1\n have h1ρ : (1 - ρ) ≠ 0 := ne_of_gt (sub_pos.mpr hρ1)\n have hπsum' : ∑' i, π 0 * ρ ^ i = 1 := by\n rw [← hπsum]; exact tsum_congr (fun i => (hpow i).symm)\n rw [tsum_mul_left, hgeo] at hπsum'\n have hπ0eq : π 0 = 1 - ρ := by\n field_simp at hπsum'\n linarith [hπsum']\n intro i\n rw [hpow i, hπ0eq]", "time_seconds": 90, "topic": "Continuous-time Markov & queues" }, { "question_id": 31, "problem_name": "stochastic_processes_submartingale_convergence_theorem", "informal_statement": "Let (Z_n) be a submartingale satisfying E[|Z_n|] ≤ M < ∞ for all n.\n\nShow that Z_n converges with probability 1 to a finite limiting random variable.", "lean_statement": "theorem q31_submartingale_convergence (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (Z : ℕ → Ω → ℝ) (hZ : Submartingale Z ℱ μ)\n (M : ℝ) (hbdd : ∀ n, ∫ ω, |Z n ω| ∂μ ≤ M) :\n ∃ Zinf : Ω → ℝ, ∀ᵐ ω ∂μ, Tendsto (fun n => Z n ω) atTop (𝓝 (Zinf ω)) := by\n sorry", "true_to_source": "literal", "proof": "theorem q31_submartingale_convergence (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (Z : ℕ → Ω → ℝ) (hZ : Submartingale Z ℱ μ)\n (M : ℝ) (hbdd : ∀ n, ∫ ω, |Z n ω| ∂μ ≤ M) :\n ∃ Zinf : Ω → ℝ, ∀ᵐ ω ∂μ, Tendsto (fun n => Z n ω) atTop (𝓝 (Zinf ω)) := by\n refine ⟨ℱ.limitProcess Z μ, ?_⟩\n have hbound : ∀ n, eLpNorm (Z n) 1 μ ≤ (M.toNNReal : ℝ≥0∞) := by\n intro n\n rw [eLpNorm_one_eq_lintegral_enorm,\n ← ofReal_integral_norm_eq_lintegral_enorm (hZ.integrable n)]\n have hle : (∫ x, ‖Z n x‖ ∂μ) ≤ M := by\n simpa [Real.norm_eq_abs] using hbdd n\n calc ENNReal.ofReal (∫ x, ‖Z n x‖ ∂μ)\n ≤ ENNReal.ofReal M := ENNReal.ofReal_le_ofReal hle\n _ = (M.toNNReal : ℝ≥0∞) := rfl\n exact hZ.ae_tendsto_limitProcess hbound", "time_seconds": 115, "topic": "Martingales & stopping" }, { "question_id": 32, "problem_name": "stochastic_processes_markov_modulated_matrix_generating", "informal_statement": "Let a Markov-modulated random walk have modulating chain with transition matrix [P] and per-transition increment generating functions g_{ij}(r), and define [A(r)] by A_{ij}(r) = P_{ij} g_{ij}(r).\n\nShow that E[e^{r S_n}] is obtained from the nth power of [A(r)], so that the growth rate of the walk's moment generating function is governed by the largest eigenvalue of [A(r)].", "lean_statement": "theorem q32_markov_modulated_matrix_generating {σ : Type*} [Fintype σ] [DecidableEq σ]\n (P : Matrix σ σ ℝ) (g : σ → σ → ℝ)\n (A : Matrix σ σ ℝ) (hA : ∀ i j, A i j = P i j * g i j)\n (Φ : ℕ → σ → ℝ) (hΦ0 : ∀ i, Φ 0 i = 1)\n (hΦrec : ∀ n i, Φ (n + 1) i = ∑ j, A i j * Φ n j) :\n ∀ n i, Φ n i = ∑ j, (A ^ n) i j := by\n sorry", "true_to_source": "abstract", "proof": "theorem q32_markov_modulated_matrix_generating {σ : Type*} [Fintype σ] [DecidableEq σ]\n (P : Matrix σ σ ℝ) (g : σ → σ → ℝ)\n (A : Matrix σ σ ℝ) (hA : ∀ i j, A i j = P i j * g i j)\n (Φ : ℕ → σ → ℝ) (hΦ0 : ∀ i, Φ 0 i = 1)\n (hΦrec : ∀ n i, Φ (n + 1) i = ∑ j, A i j * Φ n j) :\n ∀ n i, Φ n i = ∑ j, (A ^ n) i j := by\n intro n\n induction n with\n | zero => intro i; simp [hΦ0, Matrix.one_apply]\n | succ k ih =>\n intro i\n rw [hΦrec, pow_succ']\n simp only [ih, Finset.mul_sum, Matrix.mul_apply]\n rw [Finset.sum_comm]", "time_seconds": 124, "topic": "Random walks & large deviations" }, { "question_id": 33, "problem_name": "stochastic_processes_brownian_motion_martingale", "informal_statement": "Let B be a standard Brownian motion with its natural filtration.\n\nShow that B(t) is a martingale.", "lean_statement": "theorem q33_brownian_motion_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℝ m0) (B : ℝ → Ω → ℝ)\n (hadap : Adapted ℱ B)\n (hint : ∀ t, Integrable (B t) μ)\n (hB0 : ∀ᵐ ω ∂μ, B 0 ω = 0)\n (hincr : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n HasLaw (fun ω => B t ω - B s ω) (gaussianReal 0 (t - s).toNNReal) μ)\n (hpast : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n Indep (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (ℱ s) μ) :\n ∀ s t : ℝ, 0 ≤ s → s ≤ t → μ[B t | (ℱ s)] =ᵐ[μ] B s := by\n sorry", "true_to_source": "abstract", "proof": "theorem q33_brownian_motion_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℝ m0) (B : ℝ → Ω → ℝ)\n (hadap : Adapted ℱ B)\n (hint : ∀ t, Integrable (B t) μ)\n (hB0 : ∀ᵐ ω ∂μ, B 0 ω = 0)\n (hincr : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n HasLaw (fun ω => B t ω - B s ω) (gaussianReal 0 (t - s).toNNReal) μ)\n (hpast : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n Indep (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (ℱ s) μ) :\n ∀ s t : ℝ, 0 ≤ s → s ≤ t → μ[B t | (ℱ s)] =ᵐ[μ] B s := by\n intro s t hs hst\n have hBt : Measurable (B t) := (hadap t).mono (ℱ.le t) le_rfl\n have hBs : Measurable (B s) := (hadap s).mono (ℱ.le s) le_rfl\n have hg_meas : Measurable (fun ω => B t ω - B s ω) := hBt.sub hBs\n have hg_int : Integrable (fun ω => B t ω - B s ω) μ := (hint t).sub (hint s)\n have hindep := hpast s t hs hst\n have hle1 : MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance ≤ m0 :=\n hg_meas.comap_le\n have hle2 : ℱ s ≤ m0 := ℱ.le s\n have hsm : StronglyMeasurable[MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance]\n (fun ω => B t ω - B s ω) :=\n (measurable_iff_comap_le.2 le_rfl).stronglyMeasurable\n have hEg : μ[fun ω => B t ω - B s ω] = 0 := by\n rw [(hincr s t hs hst).integral_eq, integral_id_gaussianReal]\n have hcond_g : μ[fun ω => B t ω - B s ω | ℱ s] =ᵐ[μ] fun _ => (0 : ℝ) := by\n have h := condExp_indep_eq hle1 hle2 hsm hindep\n rw [hEg] at h\n exact h\n have hcond_Bs : μ[B s | ℱ s] = B s :=\n condExp_of_stronglyMeasurable hle2 (hadap s).stronglyMeasurable (hint s)\n have key : (B t) = (B s) + (fun ω => B t ω - B s ω) := by\n funext ω; show B t ω = B s ω + (B t ω - B s ω); ring\n have hadd0 : μ[(B s) + (fun ω => B t ω - B s ω) | ℱ s] =ᵐ[μ]\n (μ[B s | ℱ s] + μ[fun ω => B t ω - B s ω | ℱ s]) :=\n condExp_add (hint s) hg_int _\n have hadd : μ[B t | ℱ s] =ᵐ[μ]\n (μ[B s | ℱ s] + μ[fun ω => B t ω - B s ω | ℱ s]) := by\n calc μ[B t | ℱ s] = μ[(B s) + (fun ω => B t ω - B s ω) | ℱ s] :=\n congrArg (fun h => μ[h | ℱ s]) key\n _ =ᵐ[μ] _ := hadd0\n filter_upwards [hadd, hcond_g] with ω h1 h3\n rw [h1]\n simp [Pi.add_apply, hcond_Bs, h3]", "time_seconds": 474, "topic": "Brownian motion & stochastic calculus" }, { "question_id": 38, "problem_name": "stochastic_processes_markov_chain_1", "informal_statement": "A company issues N different types of coupons. Each coupon is equally likely to be each of the N types.\n A collector desires a complete set. Let S represent the sum of (1/k), for 1 ≤ k ≤ N.\n \n Show that the expected time for the collector to obtain all N types is N*S.", "lean_statement": "theorem q38_coupon_collector_expected (N : ℕ) (hN : 0 < N) (e : ℕ → ℝ) (hbN : e N = 0)\n (hrec : ∀ k, k < N →\n e k = 1 + ((k : ℝ) / N) * e k + (((N : ℝ) - k) / N) * e (k + 1)) :\n e 0 = (N : ℝ) * ∑ k ∈ Finset.Icc 1 N, (1 : ℝ) / k := by sorry", "true_to_source": "abstract", "proof": "theorem q38_coupon_collector_expected (N : ℕ) (hN : 0 < N) (e : ℕ → ℝ) (hbN : e N = 0)\n (hrec : ∀ k, k < N →\n e k = 1 + ((k : ℝ) / N) * e k + (((N : ℝ) - k) / N) * e (k + 1)) :\n e 0 = (N : ℝ) * ∑ k ∈ Finset.Icc 1 N, (1 : ℝ) / k := by\n have hNR : (0:ℝ) < N := by exact_mod_cast hN\n have hNne : (N:ℝ) ≠ 0 := ne_of_gt hNR\n have step : ∀ k, k < N → e k - e (k+1) = (N:ℝ)/((N:ℝ) - (k:ℝ)) := by\n intro k hk\n have hkR : (k:ℝ) < (N:ℝ) := by exact_mod_cast hk\n have hane : (N:ℝ) - (k:ℝ) ≠ 0 := sub_ne_zero.mpr (ne_of_gt hkR)\n have hh := hrec k hk\n rw [eq_div_iff hane]\n field_simp at hh\n nlinarith [hh]\n have tele : ∑ i ∈ Finset.range N, (e i - e (i+1)) = e 0 - e N :=\n Finset.sum_range_sub' e N\n have reindex : ∑ i ∈ Finset.range N, (1:ℝ)/((N:ℝ)-i) = ∑ k ∈ Finset.Icc 1 N, (1:ℝ)/k := by\n rw [← Finset.Ico_add_one_right_eq_Icc 1 N, Finset.sum_Ico_eq_sum_range]\n simp only [Nat.add_sub_cancel]\n rw [← Finset.sum_range_reflect (fun i => (1:ℝ)/((N:ℝ)-i)) N]\n apply Finset.sum_congr rfl\n intro i hi\n rw [Finset.mem_range] at hi\n have hns : (N - 1 - i : ℕ) = N - (i+1) := by omega\n rw [hns]\n have hcast : ((N - (i+1) : ℕ) : ℝ) = (N:ℝ) - (i+1) := by\n have hle : i + 1 ≤ N := by omega\n push_cast [Nat.cast_sub hle]; ring\n rw [show (N:ℝ) - ((N - (i+1) : ℕ):ℝ) = (i+1) by rw [hcast]; ring]\n push_cast; ring\n have h1 : e 0 = ∑ i ∈ Finset.range N, (e i - e (i+1)) := by rw [tele, hbN, sub_zero]\n have h2 : ∑ i ∈ Finset.range N, (e i - e (i+1))\n = ∑ i ∈ Finset.range N, (N:ℝ)/((N:ℝ)-(i:ℝ)) := by\n apply Finset.sum_congr rfl\n intro i hi\n rw [Finset.mem_range] at hi\n exact step i hi\n rw [h1, h2, show (∑ i ∈ Finset.range N, (N:ℝ)/((N:ℝ)-(i:ℝ)))\n = (N:ℝ) * ∑ i ∈ Finset.range N, (1:ℝ)/((N:ℝ)-(i:ℝ)) by\n rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro i _; rw [mul_one_div], reindex]", "time_seconds": 928, "topic": "Markov chains (finite & countable)" }, { "question_id": 40, "problem_name": "stochastic_processes_class_same_period", "informal_statement": "Let (X_n) be a finite-state Markov chain and let d(i) = gcd{n ≥ 1 : P_{ii}^n > 0} be the period of state i.\n\nShow that all states in the same communicating class have the same period.", "lean_statement": "theorem q40_class_same_period {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) :\n ∀ x y : Ω, ((∃ m : ℕ, 0 < (P ^ m) x y) ∧ (∃ k : ℕ, 0 < (P ^ k) y x)) →\n ∀ d : ℕ, (∀ n, 0 < (P ^ n) x x → d ∣ n) ↔ (∀ n, 0 < (P ^ n) y y → d ∣ n) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q40_class_same_period {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) :\n ∀ x y : Ω, ((∃ m : ℕ, 0 < (P ^ m) x y) ∧ (∃ k : ℕ, 0 < (P ^ k) y x)) →\n ∀ d : ℕ, (∀ n, 0 < (P ^ n) x x → d ∣ n) ↔ (∀ n, 0 < (P ^ n) y y → d ∣ n) := by\n have hPnn : ∀ (t : ℕ) (u v : Ω), 0 ≤ (P ^ t) u v := by\n intro t\n induction t with\n | zero => intro u v; simp only [pow_zero, Matrix.one_apply]; split <;> norm_num\n | succ s ih =>\n intro u v\n rw [pow_succ, Matrix.mul_apply]\n exact Finset.sum_nonneg (fun i _ => mul_nonneg (ih u i) (hP.1 i v))\n have step : ∀ (a b : ℕ) (u v w : Ω), (P ^ a) u v * (P ^ b) v w ≤ (P ^ (a + b)) u w := by\n intro a b u v w\n rw [pow_add, Matrix.mul_apply]\n apply Finset.single_le_sum (f := fun i => (P ^ a) u i * (P ^ b) i w)\n · intro i _; exact mul_nonneg (hPnn a u i) (hPnn b i w)\n · exact Finset.mem_univ v\n intro x y hxy d\n obtain ⟨⟨m, hm⟩, ⟨k, hk⟩⟩ := hxy\n have key : ∀ (a b : Ω) (i j : ℕ), 0 < (P ^ i) a b → 0 < (P ^ j) b a →\n (∀ n, 0 < (P ^ n) a a → d ∣ n) → (∀ n, 0 < (P ^ n) b b → d ∣ n) := by\n intro a b i j hi hj ha n hn\n have h1 : 0 < (P ^ (i + j)) a a := lt_of_lt_of_le (mul_pos hi hj) (step i j a b a)\n have h2 : 0 < (P ^ (i + n + j)) a a := by\n have hbc : 0 < (P ^ (n + j)) b a := lt_of_lt_of_le (mul_pos hn hj) (step n j b b a)\n have h3 : 0 < (P ^ (i + (n + j))) a a := lt_of_lt_of_le (mul_pos hi hbc) (step i (n + j) a b a)\n have e : i + (n + j) = i + n + j := by omega\n rwa [e] at h3\n have d1 : d ∣ (i + j) := ha _ h1\n have d2 : d ∣ (i + n + j) := ha _ h2\n have e2 : i + n + j = (i + j) + n := by omega\n rw [e2] at d2\n exact (Nat.dvd_add_right d1).mp d2\n constructor\n · exact key x y m k hm hk\n · exact key y x k m hk hm", "time_seconds": 157, "topic": "Markov chains (finite & countable)" }, { "question_id": 41, "problem_name": "stochastic_processes_mdp_optimal_policy_finite_horizon", "informal_statement": "Consider a finite-state, finite-decision Markov decision problem over a horizon of n transitions.\n\nShow that the dynamic programming recursion yields an optimal policy, and that the decision achieving the maximum at each stage depends only on the current state and the number of stages remaining.", "lean_statement": "theorem q41_optimal_policy_finite_horizon {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n {K : Type*} [Fintype K] [Nonempty K]\n (r : K → Ω → ℝ) (P : K → Matrix Ω Ω ℝ) (hP : ∀ k, IsStochastic (P k))\n (v : ℕ → Ω → ℝ) (hv0 : ∀ i, v 0 i = 0)\n (hvrec : ∀ n i, v (n + 1) i = ⨆ k, (r k i + ∑ j, P k i j * v n j)) :\n ∃ μ : ℕ → Ω → K, ∀ n i,\n r (μ n i) i + ∑ j, P (μ n i) i j * v n j = ⨆ k, (r k i + ∑ j, P k i j * v n j) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q41_optimal_policy_finite_horizon {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n {K : Type*} [Fintype K] [Nonempty K]\n (r : K → Ω → ℝ) (P : K → Matrix Ω Ω ℝ) (hP : ∀ k, IsStochastic (P k))\n (v : ℕ → Ω → ℝ) (hv0 : ∀ i, v 0 i = 0)\n (hvrec : ∀ n i, v (n + 1) i = ⨆ k, (r k i + ∑ j, P k i j * v n j)) :\n ∃ μ : ℕ → Ω → K, ∀ n i,\n r (μ n i) i + ∑ j, P (μ n i) i j * v n j = ⨆ k, (r k i + ∑ j, P k i j * v n j) := by\n have key : ∀ n i, ∃ k, r k i + ∑ j, P k i j * v n j\n = ⨆ k, (r k i + ∑ j, P k i j * v n j) := by\n intro n i\n obtain ⟨k₀, hk₀⟩ := Finite.exists_max (fun k => r k i + ∑ j, P k i j * v n j)\n refine ⟨k₀, le_antisymm ?_ (ciSup_le hk₀)⟩\n exact le_ciSup (f := fun k => r k i + ∑ j, P k i j * v n j) (Finite.bddAbove_range _) k₀\n exact ⟨fun n i => (key n i).choose, fun n i => (key n i).choose_spec⟩", "time_seconds": 109, "topic": "Markov chains (finite & countable)" }, { "question_id": 43, "problem_name": "stochastic_processes_exponential_tilting", "informal_statement": "Let X have moment generating function M and let θ_0 be such that M(θ_0) < ∞. Define the tilted distribution by P(X_{θ_0} ≤ z) = M(θ_0)^{-1} ∫_{−∞}^z e^{θ_0 x} dP(x).\n\nShow that this defines a probability distribution and that E[X_{θ_0}] = Ṁ(θ_0)/M(θ_0).", "lean_statement": "theorem q43_exponential_tilting (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : Ω → ℝ) (hX : Measurable X) (θ₀ : ℝ)\n (hInt : ∀ᶠ s in nhds θ₀, Integrable (fun ω => Real.exp (s * X ω)) μ) :\n IsProbabilityMeasure (μ.tilted (fun ω => θ₀ * X ω)) ∧\n ∫ ω, X ω ∂(μ.tilted (fun ω => θ₀ * X ω))\n = deriv (fun t => mgf X μ t) θ₀ / mgf X μ θ₀ := by sorry", "true_to_source": "literal", "proof": "theorem q43_exponential_tilting (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : Ω → ℝ) (hX : Measurable X) (θ₀ : ℝ)\n (hInt : ∀ᶠ s in nhds θ₀, Integrable (fun ω => Real.exp (s * X ω)) μ) :\n IsProbabilityMeasure (μ.tilted (fun ω => θ₀ * X ω)) ∧\n ∫ ω, X ω ∂(μ.tilted (fun ω => θ₀ * X ω))\n = deriv (fun t => mgf X μ t) θ₀ / mgf X μ θ₀ := by\n have hInt0 : Integrable (fun ω => Real.exp (θ₀ * X ω)) μ := hInt.self_of_nhds\n refine ⟨isProbabilityMeasure_tilted hInt0, ?_⟩\n have hmem : θ₀ ∈ interior (integrableExpSet X μ) := by\n rw [mem_interior_iff_mem_nhds]\n exact hInt\n have hderiv : deriv (fun t => mgf X μ t) θ₀ = ∫ ω, X ω * Real.exp (θ₀ * X ω) ∂μ :=\n deriv_mgf hmem\n have hmgf : mgf X μ θ₀ = ∫ ω, Real.exp (θ₀ * X ω) ∂μ := rfl\n rw [integral_tilted, hderiv, hmgf, ← integral_div]\n refine integral_congr_ae (Filter.Eventually.of_forall (fun ω => ?_))\n simp only [smul_eq_mul]\n ring", "time_seconds": 443, "topic": "Random walks & large deviations" }, { "question_id": 51, "problem_name": "stochastic_processes_brownian_marginal", "informal_statement": "Let B be standard Brownian motion: B(0) = 0 and the increment B(t) − B(s) is normal with mean 0 and variance t − s for 0 ≤ s ≤ t. Show that B(t) is normally distributed with mean 0 and variance t.", "lean_statement": "theorem q51_brownian_marginal (μ : Measure Ω) [IsProbabilityMeasure μ]\n (B : ℝ → Ω → ℝ) (hB0 : ∀ᵐ ω ∂μ, B 0 ω = 0)\n (hincr : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n HasLaw (fun ω => B t ω - B s ω) (gaussianReal 0 (t - s).toNNReal) μ) :\n ∀ t : ℝ, 0 ≤ t → HasLaw (B t) (gaussianReal 0 t.toNNReal) μ := by\n sorry", "true_to_source": "abstract", "proof": "theorem q51_brownian_marginal (μ : Measure Ω) [IsProbabilityMeasure μ]\n (B : ℝ → Ω → ℝ) (hB0 : ∀ᵐ ω ∂μ, B 0 ω = 0)\n (hincr : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n HasLaw (fun ω => B t ω - B s ω) (gaussianReal 0 (t - s).toNNReal) μ) :\n ∀ t : ℝ, 0 ≤ t → HasLaw (B t) (gaussianReal 0 t.toNNReal) μ := by\n intro t ht\n have hlaw := hincr 0 t le_rfl ht\n rw [sub_zero] at hlaw\n refine hlaw.congr ?_\n filter_upwards [hB0] with ω h\n rw [h, sub_zero]", "time_seconds": 52, "topic": "Brownian motion & stochastic calculus" }, { "question_id": 54, "problem_name": "stochastic_processes_continuous_mapping_theorem", "informal_statement": "Let P_n ⇒ P on a metric space S_1 and let f : S_1 → S_2 be continuous.\n\nShow that P_n f^{-1} ⇒ P f^{-1}.", "lean_statement": "theorem q54_continuous_mapping_theorem\n {S₁ S₂ : Type*} [MeasurableSpace S₁] [TopologicalSpace S₁] [OpensMeasurableSpace S₁]\n [MeasurableSpace S₂] [TopologicalSpace S₂] [BorelSpace S₂]\n (Ps : ℕ → ProbabilityMeasure S₁) (P : ProbabilityMeasure S₁)\n (hconv : Tendsto Ps atTop (𝓝 P))\n (f : S₁ → S₂) (hf : Continuous f) :\n Tendsto (fun n => (Ps n).map hf.measurable.aemeasurable) atTop\n (𝓝 (P.map hf.measurable.aemeasurable)) := by\n sorry", "true_to_source": "literal", "proof": "theorem q54_continuous_mapping_theorem\n {S₁ S₂ : Type*} [MeasurableSpace S₁] [TopologicalSpace S₁] [OpensMeasurableSpace S₁]\n [MeasurableSpace S₂] [TopologicalSpace S₂] [BorelSpace S₂]\n (Ps : ℕ → ProbabilityMeasure S₁) (P : ProbabilityMeasure S₁)\n (hconv : Tendsto Ps atTop (𝓝 P))\n (f : S₁ → S₂) (hf : Continuous f) :\n Tendsto (fun n => (Ps n).map hf.measurable.aemeasurable) atTop\n (𝓝 (P.map hf.measurable.aemeasurable)) := by\n exact ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous Ps P hconv hf", "time_seconds": 88, "topic": "Weak convergence & functional limits" }, { "question_id": 57, "problem_name": "stochastic_processes_doob_martingale", "informal_statement": "Let Z be a random variable with E[|Z|] < ∞, let (X_n) be an arbitrary sequence of random variables, and define Z_n = E[Z | X_n, X_{n−1}, …, X_1].\n\nShow that (Z_n) is a martingale (Doob's martingale).", "lean_statement": "theorem q57_doob_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (Z : Ω → ℝ) (hZ : Integrable Z μ)\n (X : ℕ → Ω → ℝ) (hX : ∀ i, StronglyMeasurable (X i)) :\n Martingale (fun n => μ[Z | (Filtration.natural X hX) n]) (Filtration.natural X hX) μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q57_doob_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (Z : Ω → ℝ) (hZ : Integrable Z μ)\n (X : ℕ → Ω → ℝ) (hX : ∀ i, StronglyMeasurable (X i)) :\n Martingale (fun n => μ[Z | (Filtration.natural X hX) n]) (Filtration.natural X hX) μ := by\n exact martingale_condExp Z (Filtration.natural X hX) μ", "time_seconds": 52, "topic": "Martingales & stopping" }, { "question_id": 60, "problem_name": "stochastic_processes_martingale_stopping_time_2", "informal_statement": "Let (X_n)_{n ≥ 0} be a martingale, and let T be a stopping time.\n\n Show that X^T is also a martingale.", "lean_statement": "theorem q60_stopped_martingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Martingale X ℱ μ) (T : Ω → ℕ)\n (hT : IsStoppingTime ℱ (natStop T)) :\n Martingale (stoppedProcess X (natStop T)) ℱ μ := by sorry", "true_to_source": "literal", "proof": "theorem q60_stopped_martingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Martingale X ℱ μ) (T : Ω → ℕ)\n (hT : IsStoppingTime ℱ (natStop T)) :\n Martingale (stoppedProcess X (natStop T)) ℱ μ := by\n rw [martingale_iff]\n refine ⟨?_, hX.submartingale.stoppedProcess hT⟩\n have hsub : Submartingale (stoppedProcess (-X) (natStop T)) ℱ μ :=\n hX.neg.submartingale.stoppedProcess hT\n have hkey : -(stoppedProcess (-X) (natStop T)) = stoppedProcess X (natStop T) := by\n funext i ω\n simp [stoppedProcess, Pi.neg_apply]\n have := hsub.neg\n rwa [hkey] at this", "time_seconds": 93, "topic": "Martingales & stopping" }, { "question_id": 63, "problem_name": "stochastic_processes_product_form_unit_expectation", "informal_statement": "Let (S_n) be a random walk with i.i.d. steps X and semi-invariant generating function γ(r) = ln E[e^{rX}], and let Z_n = exp(r S_n − n γ(r)).\n\nShow that E[Z_n] = 1 for all n.", "lean_statement": "theorem q63_product_form_unit_expectation (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (r : ℝ) (hint : Integrable (fun ω => Real.exp (r * X 0 ω)) μ)\n (γ : ℝ) (hγ : γ = cgf (X 0) μ r)\n (S : ℕ → Ω → ℝ) (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, X i ω)\n (Z : ℕ → Ω → ℝ) (hZ : ∀ n ω, Z n ω = Real.exp (r * S n ω - (n : ℝ) * γ)) (n : ℕ) :\n ∫ ω, Z n ω ∂μ = 1 := by\n sorry", "true_to_source": "abstract", "proof": "theorem q63_product_form_unit_expectation (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (r : ℝ) (hint : Integrable (fun ω => Real.exp (r * X 0 ω)) μ)\n (γ : ℝ) (hγ : γ = cgf (X 0) μ r)\n (S : ℕ → Ω → ℝ) (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, X i ω)\n (Z : ℕ → Ω → ℝ) (hZ : ∀ n ω, Z n ω = Real.exp (r * S n ω - (n : ℝ) * γ)) (n : ℕ) :\n ∫ ω, Z n ω ∂μ = 1 := by\n have hZeq : ∀ ω, Z n ω = Real.exp (r * S n ω) * Real.exp (-((n : ℝ) * γ)) := by\n intro ω; rw [hZ n ω, sub_eq_add_neg, Real.exp_add]\n have heach : ∀ i ∈ Finset.range n, mgf (X i) μ r = Real.exp γ := by\n intro i _\n rw [mgf_congr_of_identDistrib (X i) (X 0) (hident i) r, hγ, ← exp_cgf hint]\n have hmgfS : mgf (S n) μ r = (Real.exp γ) ^ n := by\n have hSeq : S n = ∑ i ∈ Finset.range n, X i := by\n funext ω; rw [hS n ω]; exact (Finset.sum_apply ω _ X).symm\n rw [hSeq, hindep.mgf_sum₀ (fun i => (hident i).aemeasurable_fst),\n Finset.prod_congr rfl heach, Finset.prod_const, Finset.card_range]\n calc ∫ ω, Z n ω ∂μ\n = ∫ ω, Real.exp (r * S n ω) * Real.exp (-((n : ℝ) * γ)) ∂μ := by\n simp only [hZeq]\n _ = (∫ ω, Real.exp (r * S n ω) ∂μ) * Real.exp (-((n : ℝ) * γ)) := integral_mul_const _ _\n _ = mgf (S n) μ r * Real.exp (-((n : ℝ) * γ)) := rfl\n _ = (Real.exp γ) ^ n * Real.exp (-((n : ℝ) * γ)) := by rw [hmgfS]\n _ = 1 := by rw [← Real.exp_nat_mul, ← Real.exp_add]; simp", "time_seconds": 273, "topic": "Random walks & large deviations" }, { "question_id": 68, "problem_name": "stochastic_processes_slln_via_backward_martingale", "informal_statement": "Let ξ_1,ξ_2,… be i.i.d. with E|ξ_1| < ∞ and let S_n = Σ_{i=1}^n ξ_i.\n\nShow, using the backward martingale convergence theorem, that S_n/n → E[ξ_1] almost surely.", "lean_statement": "theorem q68_slln_via_backward_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ξ : ℕ → Ω → ℝ) (hindep : iIndepFun ξ μ) (hident : ∀ i, IdentDistrib (ξ i) (ξ 0) μ μ)\n (hint : Integrable (ξ 0) μ)\n (S : ℕ → Ω → ℝ) (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, ξ i ω) :\n ∀ᵐ ω ∂μ, Tendsto (fun n => S n ω / (n : ℝ)) atTop (𝓝 (∫ ω, ξ 0 ω ∂μ)) := by\n sorry", "true_to_source": "literal", "proof": "theorem q68_slln_via_backward_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ξ : ℕ → Ω → ℝ) (hindep : iIndepFun ξ μ) (hident : ∀ i, IdentDistrib (ξ i) (ξ 0) μ μ)\n (hint : Integrable (ξ 0) μ)\n (S : ℕ → Ω → ℝ) (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, ξ i ω) :\n ∀ᵐ ω ∂μ, Tendsto (fun n => S n ω / (n : ℝ)) atTop (𝓝 (∫ ω, ξ 0 ω ∂μ)) := by\n have hpair : Pairwise (Function.onFun (fun x1 x2 => IndepFun x1 x2 μ) ξ) := by\n intro i j hij\n exact hindep.indepFun hij\n have hlaw := ProbabilityTheory.strong_law_ae_real ξ hint hpair hident\n filter_upwards [hlaw] with ω hω\n simp only [hS]\n exact hω", "time_seconds": 165, "topic": "Martingales & stopping" }, { "question_id": 69, "problem_name": "stochastic_processes_ctmc_steady_state_process", "informal_statement": "Let (X(t)) be an irreducible continuous-time Markov chain whose embedded chain is positive-recurrent with stationary distribution (π_i), and whose transition rates ν_i satisfy Σ_i π_i/ν_i < ∞.\n\nShow that the limiting process probabilities are p_j = (π_j/ν_j) / Σ_k (π_k/ν_k).", "lean_statement": "theorem q69_ctmc_steady_state_process\n {S : Type*} [Countable S]\n (P : S → S → ℝ) (hP0 : ∀ i j, 0 ≤ P i j) (hPstoch : ∀ i, ∑' j, P i j = 1)\n (hPloop : ∀ i, P i i = 0)\n (π : S → ℝ) (hπpos : ∀ i, 0 < π i) (hπsum : ∑' i, π i = 1)\n (hπstat : ∀ j, ∑' i, π i * P i j = π j)\n (ν : S → ℝ) (hν : ∀ i, 0 < ν i)\n (hsum : Summable (fun k => π k / ν k))\n (p : S → ℝ) (hp : ∀ j, p j = (π j / ν j) / (∑' k, π k / ν k)) :\n (∀ j, 0 ≤ p j) ∧ (∑' j, p j = 1) ∧\n (∀ j, p j * ν j = ∑' i, p i * (ν i * P i j)) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q69_ctmc_steady_state_process\n {S : Type*} [Countable S]\n (P : S → S → ℝ) (hP0 : ∀ i j, 0 ≤ P i j) (hPstoch : ∀ i, ∑' j, P i j = 1)\n (hPloop : ∀ i, P i i = 0)\n (π : S → ℝ) (hπpos : ∀ i, 0 < π i) (hπsum : ∑' i, π i = 1)\n (hπstat : ∀ j, ∑' i, π i * P i j = π j)\n (ν : S → ℝ) (hν : ∀ i, 0 < ν i)\n (hsum : Summable (fun k => π k / ν k))\n (p : S → ℝ) (hp : ∀ j, p j = (π j / ν j) / (∑' k, π k / ν k)) :\n (∀ j, 0 ≤ p j) ∧ (∑' j, p j = 1) ∧\n (∀ j, p j * ν j = ∑' i, p i * (ν i * P i j)) := by\n have hne : Nonempty S := by\n rcases isEmpty_or_nonempty S with h | h\n · exfalso\n have hz : π = fun _ => (0:ℝ) := funext (fun i => (h.false i).elim)\n rw [hz, tsum_zero] at hπsum; norm_num at hπsum\n · exact h\n have hZpos : 0 < ∑' k, π k / ν k := by\n obtain ⟨k0⟩ := hne\n have h1 : π k0 / ν k0 ≤ ∑' k, π k / ν k :=\n hsum.le_tsum k0 (fun j _ => div_nonneg (hπpos j).le (hν j).le)\n exact lt_of_lt_of_le (div_pos (hπpos k0) (hν k0)) h1\n have hZ : (∑' k, π k / ν k) ≠ 0 := hZpos.ne'\n refine ⟨?_, ?_, ?_⟩\n · intro j\n rw [hp j]\n exact div_nonneg (div_nonneg (hπpos j).le (hν j).le) hZpos.le\n · rw [tsum_congr hp, tsum_div_const, div_self hZ]\n · intro j\n have hpt : ∀ i, p i * (ν i * P i j) = π i * P i j / (∑' k, π k / ν k) := by\n intro i\n rw [hp i]\n have hνi : ν i ≠ 0 := (hν i).ne'\n field_simp\n rw [tsum_congr hpt, tsum_div_const, hπstat j, hp j]\n have hνj : ν j ≠ 0 := (hν j).ne'\n field_simp", "time_seconds": 639, "topic": "Continuous-time Markov & queues" }, { "question_id": 71, "problem_name": "stochastic_processes_stopping_times", "informal_statement": "Suppose τ and τ' are stopping times.\n \n Show that τ + τ', τ ∧ τ', and τ ∨ τ' are also stopping times.", "lean_statement": "theorem q71_stopping_times {Ω : Type*} {m : MeasurableSpace Ω}\n (ℱ : Filtration ℕ m) (τ σ : Ω → WithTop ℕ)\n (hτ : IsStoppingTime ℱ τ) (hσ : IsStoppingTime ℱ σ) :\n IsStoppingTime ℱ (fun ω => τ ω + σ ω) ∧ IsStoppingTime ℱ (τ ⊓ σ) ∧\n IsStoppingTime ℱ (τ ⊔ σ) := by sorry", "true_to_source": "literal", "proof": "theorem q71_stopping_times {Ω : Type*} {m : MeasurableSpace Ω}\n (ℱ : Filtration ℕ m) (τ σ : Ω → WithTop ℕ)\n (hτ : IsStoppingTime ℱ τ) (hσ : IsStoppingTime ℱ σ) :\n IsStoppingTime ℱ (fun ω => τ ω + σ ω) ∧ IsStoppingTime ℱ (τ ⊓ σ) ∧\n IsStoppingTime ℱ (τ ⊔ σ) :=\n ⟨hτ.add hσ, hτ.min hσ, hτ.max hσ⟩", "time_seconds": 76, "topic": "Martingales & stopping" }, { "question_id": 72, "problem_name": "stochastic_processes_submartingale_inequality_1", "informal_statement": "Let X = (X_n)_{n ≥ 0} be a non-negative submartingale.\n Let X_n^* = max_{0 ≤ k ≤ n} X_k. \n\n Show that λℙ[X_n^* ≥ λ] ≤ 𝔼[X_n * 1_{X_n^* ≥ λ}] ≤ 𝔼[X_n].", "lean_statement": "theorem q72_doob_maximal (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hX : Submartingale X ℱ μ) (hnn : ∀ n ω, 0 ≤ X n ω)\n (lam : ℝ) (hlam : 0 < lam) (n : ℕ) :\n lam * (μ {ω | lam ≤ runningMax X n ω}).toReal ≤ ∫ ω in {ω | lam ≤ runningMax X n ω}, X n ω ∂μ ∧\n ∫ ω in {ω | lam ≤ runningMax X n ω}, X n ω ∂μ ≤ ∫ ω, X n ω ∂μ := by sorry", "true_to_source": "literal", "proof": "theorem q72_doob_maximal (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hX : Submartingale X ℱ μ) (hnn : ∀ n ω, 0 ≤ X n ω)\n (lam : ℝ) (hlam : 0 < lam) (n : ℕ) :\n lam * (μ {ω | lam ≤ runningMax X n ω}).toReal ≤ ∫ ω in {ω | lam ≤ runningMax X n ω}, X n ω ∂μ ∧\n ∫ ω in {ω | lam ≤ runningMax X n ω}, X n ω ∂μ ≤ ∫ ω, X n ω ∂μ := by\n set S := {ω | lam ≤ runningMax X n ω} with hSdef\n have hnn' : (0 : ℕ → Ω → ℝ) ≤ X := fun i ω => hnn i ω\n have hrm : ∀ ω, runningMax X n ω\n = (Finset.range (n+1)).sup' Finset.nonempty_range_add_one (fun k => X k ω) := by\n intro ω\n rw [runningMax]\n apply le_antisymm\n · apply ciSup_le\n intro k\n exact Finset.le_sup' (fun k => X k ω) (Finset.mem_range.mpr k.isLt)\n · apply Finset.sup'_le\n intro k hk\n exact le_ciSup (f := fun i : Fin (n+1) => X (i : ℕ) ω)\n (Finite.bddAbove_range _) (⟨k, Finset.mem_range.mp hk⟩ : Fin (n+1))\n have hSeq : S = {ω | (lam.toNNReal : ℝ) ≤ (Finset.range (n+1)).sup'\n Finset.nonempty_range_add_one (fun k => X k ω)} := by\n rw [hSdef]; ext ω\n simp only [Set.mem_setOf_eq, hrm ω, Real.coe_toNNReal _ hlam.le]\n have hmax := maximal_ineq hX hnn' (ε := lam.toNNReal) n\n rw [← hSeq] at hmax\n have hintnn : 0 ≤ ∫ ω in S, X n ω ∂μ :=\n integral_nonneg_of_ae (ae_of_all _ (fun ω => hnn n ω))\n refine ⟨?_, ?_⟩\n · have hLne : (↑lam.toNNReal * μ S) ≠ ⊤ :=\n ENNReal.mul_ne_top (by simp) (measure_ne_top μ S)\n have hRne : ENNReal.ofReal (∫ ω in S, X n ω ∂μ) ≠ ⊤ := ENNReal.ofReal_ne_top\n have h1 := (ENNReal.toReal_le_toReal hLne hRne).mpr hmax\n rw [ENNReal.toReal_mul, ENNReal.coe_toReal, ENNReal.toReal_ofReal hintnn,\n Real.coe_toNNReal _ hlam.le] at h1\n exact h1\n · exact setIntegral_le_integral (hX.integrable n) (ae_of_all _ (fun ω => hnn n ω))", "time_seconds": 393, "topic": "Martingales & stopping" }, { "question_id": 75, "problem_name": "stochastic_processes_strong_stationary_time_1", "informal_statement": "Let (X_n)_{n ≥ 0} be an irreducible Markov chain with stationary measure π.\n Let the separation distance be S_x (n) = max_y (1 − P^n (x, y) / π(y)).\n \n Show that ||P^n (x, ·) - π||_{TV} ≤ S_x (n).", "lean_statement": "theorem q75_tv_le_sep {Ω : Type*} [Fintype Ω] [Nonempty Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (π : Ω → ℝ) (hπ : IsStationary P π)\n (hπpos : ∀ x, 0 < π x) (x : Ω) (n : ℕ) :\n tvDist (fun y => (P ^ n) x y) π ≤ ⨆ y, (1 - (P ^ n) x y / π y) := by sorry", "true_to_source": "literal", "proof": "theorem q75_tv_le_sep {Ω : Type*} [Fintype Ω] [Nonempty Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (π : Ω → ℝ) (hπ : IsStationary P π)\n (hπpos : ∀ x, 0 < π x) (x : Ω) (n : ℕ) :\n tvDist (fun y => (P ^ n) x y) π ≤ ⨆ y, (1 - (P ^ n) x y / π y) := by\n -- power of stochastic matrix has nonneg entries and rows summing to 1\n have hrow : ∀ (m : ℕ) (a : Ω), ∑ b, (P ^ m) a b = 1 := by\n intro m\n induction m with\n | zero => intro a; simp [pow_zero, Matrix.one_apply, Finset.sum_ite_eq]\n | succ k ih =>\n intro a\n rw [pow_succ]\n simp_rw [Matrix.mul_apply]\n rw [Finset.sum_comm]\n simp_rw [← Finset.mul_sum, hP.2, mul_one]\n exact ih a\n set a : Ω → ℝ := fun y => (P ^ n) x y with ha_def\n set sep : ℝ := ⨆ y, (1 - a y / π y) with hsep_def\n have hasum : ∑ y, a y = 1 := hrow n x\n have hbsum : ∑ y, π y = 1 := hπ.2.1\n have hbdd : BddAbove (Set.range (fun y => 1 - a y / π y)) :=\n (Set.finite_range _).bddAbove\n -- sep ≥ 0\n have hsep_nonneg : 0 ≤ sep := by\n have hex : ∃ y, a y ≤ π y := by\n by_contra hcon\n push_neg at hcon\n have hlt : ∑ y, π y < ∑ y, a y :=\n Finset.sum_lt_sum_of_nonempty Finset.univ_nonempty (fun y _ => hcon y)\n rw [hasum, hbsum] at hlt\n exact lt_irrefl 1 hlt\n obtain ⟨y, hy⟩ := hex\n have h1 : a y / π y ≤ 1 := (div_le_one (hπpos y)).mpr hy\n have : 0 ≤ 1 - a y / π y := by linarith\n calc (0:ℝ) ≤ 1 - a y / π y := this\n _ ≤ sep := le_ciSup hbdd y\n -- identity: ∑|a-π| = 2 ∑ max(π-a,0)\n have hidentity : ∑ y, |a y - π y| = 2 * ∑ y, max (π y - a y) 0 := by\n have e1 : ∀ y, |a y - π y| = 2 * max (π y - a y) 0 - (π y - a y) := by\n intro y\n rw [abs_sub_comm]\n rcases le_total 0 (π y - a y) with h | h\n · rw [abs_of_nonneg h, max_eq_left h]; ring\n · rw [abs_of_nonpos h, max_eq_right h]; ring\n rw [Finset.sum_congr rfl (fun y _ => e1 y), Finset.sum_sub_distrib, ← Finset.mul_sum]\n have hz : ∑ y, (π y - a y) = 0 := by\n rw [Finset.sum_sub_distrib, hbsum, hasum]; ring\n rw [hz]; ring\n -- per-term bound\n have hmax_le : ∀ y, max (π y - a y) 0 ≤ π y * sep := by\n intro y\n rcases le_total (π y - a y) 0 with h | h\n · rw [max_eq_right h]\n exact mul_nonneg (le_of_lt (hπpos y)) hsep_nonneg\n · rw [max_eq_left h]\n have hterm : (1 - a y / π y) ≤ sep := le_ciSup hbdd y\n have hne : π y ≠ 0 := ne_of_gt (hπpos y)\n have hrw : π y - a y = π y * (1 - a y / π y) := by field_simp\n rw [hrw]\n exact mul_le_mul_of_nonneg_left hterm (le_of_lt (hπpos y))\n -- assemble\n simp only [Auto.tvDist]\n calc (1/2) * ∑ y, |a y - π y|\n = (1/2) * (2 * ∑ y, max (π y - a y) 0) := by rw [hidentity]\n _ = ∑ y, max (π y - a y) 0 := by ring\n _ ≤ ∑ y, π y * sep := Finset.sum_le_sum (fun y _ => hmax_le y)\n _ = (∑ y, π y) * sep := by rw [← Finset.sum_mul]\n _ = sep := by rw [hbsum]; ring", "time_seconds": 178, "topic": "Markov chains (finite & countable)" }, { "question_id": 78, "problem_name": "stochastic_processes_random_walk_square_martingale", "informal_statement": "Let X_1,X_2,… be i.i.d. with mean 0 and variance σ^2, and let S_n = Σ_{i=1}^n X_i.\n\nShow that S_n^2 − nσ^2 is a martingale.", "lean_statement": "theorem q78_square_random_walk_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (X : ℕ → Ω → ℝ)\n (hadap : Adapted ℱ (fun n ω => ∑ i ∈ Finset.range n, X i ω))\n (hint : ∀ n, Integrable (fun ω => (X n ω) ^ 2) μ) (σ2 : ℝ)\n (hmean : ∀ n, μ[X n | ℱ n] =ᵐ[μ] fun _ => (0 : ℝ))\n (hvar : ∀ n, μ[fun ω => (X n ω) ^ 2 | ℱ n] =ᵐ[μ] fun _ => σ2) :\n Martingale (fun n ω => (∑ i ∈ Finset.range n, X i ω) ^ 2 - (n : ℝ) * σ2) ℱ μ := by\n sorry", "true_to_source": "abstract", "proof": "theorem q78_square_random_walk_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (X : ℕ → Ω → ℝ)\n (hadap : Adapted ℱ (fun n ω => ∑ i ∈ Finset.range n, X i ω))\n (hint : ∀ n, Integrable (fun ω => (X n ω) ^ 2) μ) (σ2 : ℝ)\n (hmean : ∀ n, μ[X n | ℱ n] =ᵐ[μ] fun _ => (0 : ℝ))\n (hvar : ∀ n, μ[fun ω => (X n ω) ^ 2 | ℱ n] =ᵐ[μ] fun _ => σ2) :\n Martingale (fun n ω => (∑ i ∈ Finset.range n, X i ω) ^ 2 - (n : ℝ) * σ2) ℱ μ := by\n have hSmeasble : ∀ n, Measurable[m0] (fun ω => ∑ i ∈ Finset.range n, X i ω) :=\n fun n => (hadap n).mono (ℱ.le n) le_rfl\n have hXeq : ∀ n, X n = fun ω => (∑ i ∈ Finset.range (n + 1), X i ω)\n - (∑ i ∈ Finset.range n, X i ω) := by\n intro n; funext ω; rw [Finset.sum_range_succ]; ring\n have hXmeas : ∀ n, Measurable (X n) := by\n intro n; rw [hXeq n]; exact (hSmeasble (n + 1)).sub (hSmeasble n)\n have hXaesm : ∀ n, AEStronglyMeasurable (X n) μ :=\n fun n => (hXmeas n).aestronglyMeasurable\n have hXmemLp : ∀ n, MemLp (X n) 2 μ :=\n fun n => (memLp_two_iff_integrable_sq (hXaesm n)).2 (hint n)\n have hXint : ∀ n, Integrable (X n) μ :=\n fun n => (hXmemLp n).integrable (by norm_num)\n have hSmemLp : ∀ n, MemLp (fun ω => ∑ i ∈ Finset.range n, X i ω) 2 μ :=\n fun n => memLp_finsetSum (Finset.range n) (fun i _ => hXmemLp i)\n have hS2int : ∀ n, Integrable (fun ω => (∑ i ∈ Finset.range n, X i ω) ^ 2) μ :=\n fun n => (hSmemLp n).integrable_sq\n have hSXint : ∀ n, Integrable (fun ω => (∑ i ∈ Finset.range n, X i ω) * X n ω) μ :=\n fun n => (hSmemLp n).integrable_mul (hXmemLp n)\n refine martingale_of_condExp_sub_eq_zero_nat ?_ ?_ ?_\n · intro n\n exact (((hadap n).pow_const 2).sub measurable_const).stronglyMeasurable\n · intro n\n exact (hS2int n).sub (integrable_const _)\n · intro i\n have hincrem :\n (fun n ω => (∑ k ∈ Finset.range n, X k ω) ^ 2 - (n : ℝ) * σ2) (i + 1)\n - (fun n ω => (∑ k ∈ Finset.range n, X k ω) ^ 2 - (n : ℝ) * σ2) i\n = (2 : ℝ) • ((fun ω => ∑ k ∈ Finset.range i, X k ω) * X i)\n + (fun ω => (X i ω) ^ 2) - (fun _ => σ2) := by\n funext ω\n simp only [Pi.add_apply, Pi.sub_apply, Pi.smul_apply, Pi.mul_apply, smul_eq_mul,\n Finset.sum_range_succ]\n push_cast\n ring\n rw [hincrem]\n have h2P_int : Integrable ((2 : ℝ) • ((fun ω => ∑ k ∈ Finset.range i, X k ω) * X i)) μ :=\n (hSXint i).smul (2 : ℝ)\n have hAB_int : Integrable ((2 : ℝ) • ((fun ω => ∑ k ∈ Finset.range i, X k ω) * X i)\n + (fun ω => (X i ω) ^ 2)) μ := h2P_int.add (hint i)\n have h1 := condExp_sub hAB_int (integrable_const σ2) (ℱ i)\n have h2 := condExp_add h2P_int (hint i) (ℱ i)\n have h3 := condExp_smul (μ := μ) (2 : ℝ)\n ((fun ω => ∑ k ∈ Finset.range i, X k ω) * X i) (ℱ i)\n have h4 := condExp_mul_of_stronglyMeasurable_left (m := ℱ i)\n (hadap i).stronglyMeasurable (hSXint i) (hXint i)\n have hC : μ[(fun _ : Ω => σ2) | ℱ i] = fun _ => σ2 := condExp_const (ℱ.le i) σ2\n filter_upwards [h1, h2, h3, h4, hmean i, hvar i] with ω e1 e2 e3 e4 e5 e6\n simp only [Pi.sub_apply, Pi.add_apply, Pi.smul_apply, Pi.mul_apply, smul_eq_mul,\n Pi.zero_apply, hC] at e1 e2 e3 e4 e5 e6 ⊢\n rw [e1, e2, e3, e4, e5, e6]\n ring", "time_seconds": 491, "topic": "Martingales & stopping" }, { "question_id": 81, "problem_name": "stochastic_processes_markov_modulated_product_martingale", "informal_statement": "Let a Markov-modulated random walk have matrix [A(r)] with largest eigenvalue ρ(r) and corresponding positive right eigenvector ν(r), and define Z_n = exp(r S_n) ν_{Y_n}(r) / ρ(r)^n.\n\nShow that (Z_n) is a martingale.", "lean_statement": "theorem q81_markov_modulated_product_martingale {σ : Type*} [Fintype σ]\n (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (Y : ℕ → Ω → σ) (X S : ℕ → Ω → ℝ) (r ρ : ℝ) (hρ : 0 < ρ)\n (ν : σ → ℝ) (hν : ∀ s, 0 < ν s)\n (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, X i ω) (hSadap : Adapted ℱ S)\n (Z : ℕ → Ω → ℝ) (hZ : ∀ n ω, Z n ω = Real.exp (r * S n ω) * ν (Y n ω) / ρ ^ n)\n (hadap : Adapted ℱ Z) (hint : ∀ n, Integrable (Z n) μ)\n (heig : ∀ n, μ[fun ω => Real.exp (r * X n ω) * ν (Y (n + 1) ω) | ℱ n]\n =ᵐ[μ] fun ω => ρ * ν (Y n ω)) :\n Martingale Z ℱ μ := by\n sorry", "true_to_source": "abstract", "proof": "theorem q81_markov_modulated_product_martingale {σ : Type*} [Fintype σ]\n (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (Y : ℕ → Ω → σ) (X S : ℕ → Ω → ℝ) (r ρ : ℝ) (hρ : 0 < ρ)\n (ν : σ → ℝ) (hν : ∀ s, 0 < ν s)\n (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, X i ω) (hSadap : Adapted ℱ S)\n (Z : ℕ → Ω → ℝ) (hZ : ∀ n ω, Z n ω = Real.exp (r * S n ω) * ν (Y n ω) / ρ ^ n)\n (hadap : Adapted ℱ Z) (hint : ∀ n, Integrable (Z n) μ)\n (heig : ∀ n, μ[fun ω => Real.exp (r * X n ω) * ν (Y (n + 1) ω) | ℱ n]\n =ᵐ[μ] fun ω => ρ * ν (Y n ω)) :\n Martingale Z ℱ μ := by\n have hSA : StronglyAdapted ℱ Z := fun i => (hadap i).stronglyMeasurable\n refine martingale_nat hSA hint ?_\n intro n\n set f : Ω → ℝ := fun ω => Real.exp (r * S n ω) / ρ ^ (n + 1) with hf_def\n set g : Ω → ℝ := fun ω => Real.exp (r * X n ω) * ν (Y (n + 1) ω) with hg_def\n have hgce : μ[g | ℱ n] =ᵐ[μ] fun ω => ρ * ν (Y n ω) := heig n\n -- Integrability of g via contradiction with heig\n have hg_int : Integrable g μ := by\n by_contra hni\n have h0 : μ[g | ℱ n] = 0 := condExp_of_not_integrable hni\n have hzero : (fun ω => ρ * ν (Y n ω)) =ᵐ[μ] (0 : Ω → ℝ) := by\n calc (fun ω => ρ * ν (Y n ω)) =ᵐ[μ] μ[g | ℱ n] := hgce.symm\n _ = (0 : Ω → ℝ) := h0\n have hpos : ∀ ω, ρ * ν (Y n ω) ≠ 0 := fun ω => ne_of_gt (mul_pos hρ (hν (Y n ω)))\n have hfalse : ∀ᵐ ω ∂μ, False := by\n filter_upwards [hzero] with ω hω\n exact hpos ω hω\n have hmz : μ {ω | ¬ False} = 0 := ae_iff.mp hfalse\n simp only [not_false_eq_true, Set.setOf_true, measure_univ] at hmz\n exact one_ne_zero hmz\n -- f is ℱ n-measurable\n have hf_meas : StronglyMeasurable[ℱ n] f := by\n rw [hf_def]\n apply Measurable.stronglyMeasurable\n apply Measurable.div_const\n exact Real.measurable_exp.comp (measurable_const.mul (hSadap n))\n -- Z (n+1) = f * g\n have hZeq : Z (n + 1) = f * g := by\n funext ω\n simp only [Pi.mul_apply, hf_def, hg_def, hZ (n + 1) ω]\n have hSs : S (n + 1) ω = S n ω + X n ω := by\n rw [hS (n + 1) ω, hS n ω, Finset.sum_range_succ]\n rw [hSs, mul_add, Real.exp_add]\n ring\n have hfg_int : Integrable (f * g) μ := by rw [← hZeq]; exact hint (n + 1)\n have step1 : μ[Z (n + 1) | ℱ n] =ᵐ[μ] f * μ[g | ℱ n] := by\n rw [hZeq]; exact condExp_mul_of_stronglyMeasurable_left hf_meas hfg_int hg_int\n have step2 : f * μ[g | ℱ n] =ᵐ[μ] Z n := by\n filter_upwards [hgce] with ω hω\n simp only [Pi.mul_apply, hf_def]\n rw [hω, hZ n ω, pow_succ]\n have hρ0 : ρ ≠ 0 := ne_of_gt hρ\n have hρn : ρ ^ n ≠ 0 := pow_ne_zero n hρ0\n field_simp\n exact (step1.trans step2).symm", "time_seconds": 300, "topic": "Martingales & stopping" }, { "question_id": 83, "problem_name": "stochastic_processes_class_transient_recurrent", "informal_statement": "Let (X_n) be a finite-state Markov chain.\n\nShow that within any communicating class, either all states are transient or all states are recurrent.", "lean_statement": "theorem q83_class_transient_recurrent {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) :\n ∀ x y : Ω, ((∃ m : ℕ, 0 < (P ^ m) x y) ∧ (∃ k : ℕ, 0 < (P ^ k) y x)) →\n (Summable (fun n => (P ^ n) x x) ↔ Summable (fun n => (P ^ n) y y)) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q83_class_transient_recurrent {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) :\n ∀ x y : Ω, ((∃ m : ℕ, 0 < (P ^ m) x y) ∧ (∃ k : ℕ, 0 < (P ^ k) y x)) →\n (Summable (fun n => (P ^ n) x x) ↔ Summable (fun n => (P ^ n) y y)) := by\n obtain ⟨hnn, hstoch⟩ := hP\n -- entries of powers are nonnegative\n have hpow : ∀ (t : ℕ) (i j : Ω), 0 ≤ (P ^ t) i j := by\n intro t\n induction t with\n | zero => intro i j; rw [pow_zero, Matrix.one_apply]; split <;> norm_num\n | succ t ih =>\n intro i j\n rw [pow_succ, Matrix.mul_apply]\n exact Finset.sum_nonneg (fun w _ => mul_nonneg (ih i w) (hnn w j))\n -- a single path term lower-bounds the (a+b)-step probability\n have hstep : ∀ (a b : ℕ) (i z j : Ω), (P ^ a) i z * (P ^ b) z j ≤ (P ^ (a + b)) i j := by\n intro a b i z j\n rw [pow_add, Matrix.mul_apply]\n exact Finset.single_le_sum (fun w _ => mul_nonneg (hpow a i w) (hpow b w j)) (Finset.mem_univ z)\n -- one direction (symmetric)\n have solidarity : ∀ (x y : Ω) (m k : ℕ), 0 < (P ^ m) x y → 0 < (P ^ k) y x →\n Summable (fun n => (P ^ n) x x) → Summable (fun n => (P ^ n) y y) := by\n intro x y m k hm hk hx\n set c := (P ^ m) x y * (P ^ k) y x with hc\n have hcpos : 0 < c := mul_pos hm hk\n have hineq : ∀ n, c * (P ^ n) y y ≤ (P ^ (m + (n + k))) x x := by\n intro n\n calc c * (P ^ n) y y = (P ^ m) x y * ((P ^ n) y y * (P ^ k) y x) := by rw [hc]; ring\n _ ≤ (P ^ m) x y * (P ^ (n + k)) y x :=\n mul_le_mul_of_nonneg_left (hstep n k y y x) (le_of_lt hm)\n _ ≤ (P ^ (m + (n + k))) x x := hstep m (n + k) x y x\n have hginj : Function.Injective (fun n : ℕ => m + (n + k)) := by\n intro a b h; dsimp only at h; omega\n have hshift : Summable (fun n => (P ^ (m + (n + k))) x x) := hx.comp_injective hginj\n have hcy : Summable (fun n => c * (P ^ n) y y) :=\n Summable.of_nonneg_of_le\n (fun n => mul_nonneg (le_of_lt hcpos) (hpow n y y)) hineq hshift\n have h5 := hcy.mul_left c⁻¹\n simp only [inv_mul_cancel_left₀ (ne_of_gt hcpos)] at h5\n exact h5\n intro x y hxy\n obtain ⟨⟨m, hm⟩, ⟨k, hk⟩⟩ := hxy\n constructor\n · intro hx; exact solidarity x y m k hm hk hx\n · intro hy; exact solidarity y x k m hk hm hy", "time_seconds": 267, "topic": "Markov chains (finite & countable)" }, { "question_id": 85, "problem_name": "stochastic_processes_renewal_slln", "informal_statement": "Let S_n = X_1 + … + X_n where the X_i are i.i.d. with E[|X|] < ∞ and mean X̄.\n\nShow that Pr[lim_{n→∞} S_n/n = X̄] = 1.", "lean_statement": "theorem q85_renewal_slln (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hint : Integrable (X 0) μ)\n (hindep : Pairwise (fun i j => IndepFun (X i) (X j) μ))\n (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ) :\n ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => (n : ℝ)⁻¹ • (∑ i ∈ Finset.range n, X i ω))\n atTop (𝓝 (μ[X 0])) := by\n sorry", "true_to_source": "literal", "proof": "theorem q85_renewal_slln (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hint : Integrable (X 0) μ)\n (hindep : Pairwise (fun i j => IndepFun (X i) (X j) μ))\n (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ) :\n ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => (n : ℝ)⁻¹ • (∑ i ∈ Finset.range n, X i ω))\n atTop (𝓝 (μ[X 0])) := by\n exact ProbabilityTheory.strong_law_ae X hint hindep hident", "time_seconds": 48, "topic": "Weak convergence & functional limits" }, { "question_id": 86, "problem_name": "stochastic_processes_birth_death_geometric", "informal_statement": "Consider an irreducible birth-death chain on {0,1,2,…} with constant ratio ρ = p/q < 1 (up-probability p, down-probability q at every interior state).\n\nShow that the stationary distribution is π_i = (1 − ρ) ρ^i for i ≥ 0.", "lean_statement": "theorem q86_birth_death_geometric\n (P : ℕ → ℕ → ℝ) (hP0 : ∀ i j, 0 ≤ P i j) (hP1 : ∀ i, ∑' j, P i j = 1)\n (ρ : ℝ) (hρ0 : 0 < ρ) (hρ1 : ρ < 1)\n (hratio : ∀ i, P i (i + 1) = ρ * P (i + 1) i)\n (π : ℕ → ℝ) (hπ : ∀ i, π i = (1 - ρ) * ρ ^ i) :\n (∀ i, 0 ≤ π i) ∧ (∑' i, π i = 1) ∧\n (∀ i, π i * P i (i + 1) = π (i + 1) * P (i + 1) i) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q86_birth_death_geometric\n (P : ℕ → ℕ → ℝ) (hP0 : ∀ i j, 0 ≤ P i j) (hP1 : ∀ i, ∑' j, P i j = 1)\n (ρ : ℝ) (hρ0 : 0 < ρ) (hρ1 : ρ < 1)\n (hratio : ∀ i, P i (i + 1) = ρ * P (i + 1) i)\n (π : ℕ → ℝ) (hπ : ∀ i, π i = (1 - ρ) * ρ ^ i) :\n (∀ i, 0 ≤ π i) ∧ (∑' i, π i = 1) ∧\n (∀ i, π i * P i (i + 1) = π (i + 1) * P (i + 1) i) := by\n refine ⟨?_, ?_, ?_⟩\n · intro i; rw [hπ]; exact mul_nonneg (by linarith) (pow_nonneg hρ0.le i)\n · have h2 : ∑' i, π i = (1 - ρ) * ∑' i : ℕ, ρ ^ i := by\n rw [← tsum_mul_left]; exact tsum_congr (fun i => hπ i)\n rw [h2, tsum_geometric_of_lt_one hρ0.le hρ1]\n exact mul_inv_cancel₀ (by linarith)\n · intro i; rw [hπ, hπ, hratio i]; ring", "time_seconds": 94, "topic": "Markov chains (finite & countable)" }, { "question_id": 87, "problem_name": "stochastic_processes_martingale_simple_random_walk", "informal_statement": "Let X = (X_n)_{n ≥ 0} be the simple random walk on ℤ.\n\n Show that (Y_n := X_n^3 − 3nX_n)_{n ≥ 0} is a martingale.", "lean_statement": "theorem q87_srw_cubic_martingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hadap : Adapted ℱ X) (hint : ∀ n, Integrable (fun ω => (X n ω) ^ 3) μ)\n (hstep : ∀ n, ∀ᵐ ω ∂μ, X (n + 1) ω = X n ω + 1 ∨ X (n + 1) ω = X n ω - 1)\n (hsym : IsConstDrift μ ℱ X 0) :\n Martingale (fun n ω => (X n ω) ^ 3 - 3 * n * X n ω) ℱ μ := by sorry", "true_to_source": "literal", "proof": "theorem q87_srw_cubic_martingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hadap : Adapted ℱ X) (hint : ∀ n, Integrable (fun ω => (X n ω) ^ 3) μ)\n (hstep : ∀ n, ∀ᵐ ω ∂μ, X (n + 1) ω = X n ω + 1 ∨ X (n + 1) ω = X n ω - 1)\n (hsym : IsConstDrift μ ℱ X 0) :\n Martingale (fun n ω => (X n ω) ^ 3 - 3 * n * X n ω) ℱ μ := by\n have hXsm : ∀ k, StronglyMeasurable[ℱ k] (X k) := fun k => (hadap k).stronglyMeasurable\n have hbound1 : ∀ x : ℝ, |x| ≤ 1 + |x| ^ 3 := by\n intro x\n rcases le_total |x| 1 with h | h\n · nlinarith [abs_nonneg x, pow_nonneg (abs_nonneg x) 3]\n · nlinarith [abs_nonneg x, h, mul_nonneg (abs_nonneg x) (abs_nonneg x)]\n have hbound2 : ∀ x : ℝ, x ^ 2 ≤ 1 + |x| ^ 3 := by\n intro x\n rcases le_total |x| 1 with h | h\n · nlinarith [abs_nonneg x, pow_nonneg (abs_nonneg x) 3, sq_abs x]\n · nlinarith [abs_nonneg x, h, sq_abs x, mul_nonneg (abs_nonneg x) (abs_nonneg x)]\n -- integrability of X k and X k ^ 2 from integrability of X k ^ 3\n have hXint : ∀ k, Integrable (X k) μ := by\n intro k\n refine Integrable.mono' ((integrable_const (1:ℝ)).add (hint k).norm)\n ((hXsm k).mono (ℱ.le k)).aestronglyMeasurable ?_\n filter_upwards with ω\n simp only [Real.norm_eq_abs, abs_pow]\n exact hbound1 (X k ω)\n have hXsq : ∀ k, Integrable (fun ω => (X k ω) ^ 2) μ := by\n intro k\n refine Integrable.mono' ((integrable_const (1:ℝ)).add (hint k).norm)\n (((hXsm k).mono (ℱ.le k)).pow 2).aestronglyMeasurable ?_\n filter_upwards with ω\n simp only [Real.norm_eq_abs, abs_pow, Pi.add_apply, sq_abs]\n exact hbound2 (X k ω)\n refine martingale_nat ?_ ?_ ?_\n · intro n\n exact ((hXsm n).pow 3).sub ((hXsm n).const_mul (3 * (n:ℝ)))\n · intro n\n exact (hint n).sub ((hXint n).const_mul (3 * (n:ℝ)))\n · intro n\n have hg_int : Integrable (fun ω => X (n+1) ω - X n ω) μ := (hXint (n+1)).sub (hXint n)\n have hc_int : Integrable (fun ω => 3 * (X n ω)^2 - (3*(n:ℝ)+2)) μ :=\n ((hXsq n).const_mul 3).sub (integrable_const _)\n have hc_sm : StronglyMeasurable[ℱ n] (fun ω => 3 * (X n ω)^2 - (3*(n:ℝ)+2)) :=\n (((hXsm n).pow 2).const_mul 3).sub stronglyMeasurable_const\n have hYn_int : Integrable (fun ω => (X n ω)^3 - 3*(n:ℝ)*(X n ω)) μ :=\n (hint n).sub ((hXint n).const_mul (3 * (n:ℝ)))\n have hYn_sm : StronglyMeasurable[ℱ n] (fun ω => (X n ω)^3 - 3*(n:ℝ)*(X n ω)) :=\n ((hXsm n).pow 3).sub ((hXsm n).const_mul (3 * (n:ℝ)))\n have hcd_int : Integrable\n ((fun ω => 3 * (X n ω)^2 - (3*(n:ℝ)+2)) * (fun ω => X (n+1) ω - X n ω)) μ := by\n refine Integrable.mono' hc_int.norm\n ((hc_sm.mono (ℱ.le n)).aestronglyMeasurable.mul\n (((hXsm (n+1)).mono (ℱ.le (n+1))).sub ((hXsm n).mono (ℱ.le n))).aestronglyMeasurable) ?_\n filter_upwards [hstep n] with ω hω\n have hd1 : ‖X (n+1) ω - X n ω‖ = 1 := by rcases hω with h | h <;> rw [h] <;> simp\n simp only [Pi.mul_apply, norm_mul, hd1, mul_one, le_refl]\n have hidentity : (fun ω => (X (n+1) ω)^3 - (3*((n:ℝ)+1))*(X (n+1) ω)) =ᵐ[μ]\n (fun ω => 3 * (X n ω)^2 - (3*(n:ℝ)+2)) * (fun ω => X (n+1) ω - X n ω)\n + (fun ω => (X n ω)^3 - 3*(n:ℝ)*(X n ω)) := by\n filter_upwards [hstep n] with ω hω\n simp only [Pi.add_apply, Pi.mul_apply]\n rcases hω with h | h <;> rw [h] <;> ring\n have e4 : μ[(fun ω => (X n ω)^3 - 3*(n:ℝ)*(X n ω)) | ℱ n]\n =ᵐ[μ] (fun ω => (X n ω)^3 - 3*(n:ℝ)*(X n ω)) := by\n rw [condExp_of_stronglyMeasurable (ℱ.le n) hYn_sm hYn_int]\n have step : μ[(fun n ω => (X n ω)^3 - 3 * (n:ℝ) * X n ω) (n+1) | ℱ n]\n =ᵐ[μ] (fun ω => (X n ω)^3 - 3*(n:ℝ)*(X n ω)) := by\n calc μ[(fun n ω => (X n ω)^3 - 3 * (n:ℝ) * X n ω) (n+1) | ℱ n]\n =ᵐ[μ] μ[(fun ω => (X (n+1) ω)^3 - (3*((n:ℝ)+1))*(X (n+1) ω)) | ℱ n] := by\n apply condExp_congr_ae; filter_upwards with ω; push_cast; ring\n _ =ᵐ[μ] μ[(fun ω => 3 * (X n ω)^2 - (3*(n:ℝ)+2)) * (fun ω => X (n+1) ω - X n ω)\n + (fun ω => (X n ω)^3 - 3*(n:ℝ)*(X n ω)) | ℱ n] := condExp_congr_ae hidentity\n _ =ᵐ[μ] μ[(fun ω => 3 * (X n ω)^2 - (3*(n:ℝ)+2)) * (fun ω => X (n+1) ω - X n ω) | ℱ n]\n + μ[(fun ω => (X n ω)^3 - 3*(n:ℝ)*(X n ω)) | ℱ n] := condExp_add hcd_int hYn_int _\n _ =ᵐ[μ] (fun ω => 3 * (X n ω)^2 - (3*(n:ℝ)+2)) * μ[(fun ω => X (n+1) ω - X n ω) | ℱ n]\n + μ[(fun ω => (X n ω)^3 - 3*(n:ℝ)*(X n ω)) | ℱ n] :=\n EventuallyEq.add (condExp_mul_of_stronglyMeasurable_left hc_sm hcd_int hg_int)\n EventuallyEq.rfl\n _ =ᵐ[μ] (fun ω => 3 * (X n ω)^2 - (3*(n:ℝ)+2)) * (fun _ => (0:ℝ))\n + (fun ω => (X n ω)^3 - 3*(n:ℝ)*(X n ω)) :=\n EventuallyEq.add (EventuallyEq.mul EventuallyEq.rfl (hsym n)) e4\n _ =ᵐ[μ] (fun ω => (X n ω)^3 - 3*(n:ℝ)*(X n ω)) := by\n filter_upwards with ω; simp [Pi.add_apply, Pi.mul_apply]\n exact step.symm", "time_seconds": 121, "topic": "Martingales & stopping" }, { "question_id": 88, "problem_name": "stochastic_processes_stopped_submartingale_bounds", "informal_statement": "Let (Z_n) be a submartingale and let J be a stopping trial with stopped process (Z_n*).\n\nShow that E[Z_1] ≤ E[Z_n*] ≤ E[Z_n] for all n.", "lean_statement": "theorem q88_stopped_submartingale_bounds (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ]\n (Z : ℕ → Ω → ℝ) (hZ : Submartingale Z ℱ μ)\n (J : Ω → ℕ) (hJ : IsStoppingTime ℱ (natStop J)) (n : ℕ) :\n ∫ ω, Z 0 ω ∂μ ≤ ∫ ω, stoppedProcess Z (natStop J) n ω ∂μ ∧\n ∫ ω, stoppedProcess Z (natStop J) n ω ∂μ ≤ ∫ ω, Z n ω ∂μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q88_stopped_submartingale_bounds (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ]\n (Z : ℕ → Ω → ℝ) (hZ : Submartingale Z ℱ μ)\n (J : Ω → ℕ) (hJ : IsStoppingTime ℱ (natStop J)) (n : ℕ) :\n ∫ ω, Z 0 ω ∂μ ≤ ∫ ω, stoppedProcess Z (natStop J) n ω ∂μ ∧\n ∫ ω, stoppedProcess Z (natStop J) n ω ∂μ ≤ ∫ ω, Z n ω ∂μ := by\n have hP : IsStoppingTime ℱ (fun ω => min (↑n : ℕ∞) (natStop J ω)) :=\n (isStoppingTime_const ℱ n).min hJ\n have h0 : IsStoppingTime ℱ (fun _ : Ω => ((0 : ℕ) : ℕ∞)) := isStoppingTime_const ℱ 0\n have hn : IsStoppingTime ℱ (fun _ : Ω => ((n : ℕ) : ℕ∞)) := isStoppingTime_const ℱ n\n refine ⟨?_, ?_⟩\n · have h := hZ.expected_stoppedValue_mono h0 hP\n (fun ω => zero_le) (N := n) (fun ω => min_le_left _ _)\n simpa only [stoppedValue_const, stoppedProcess_eq_stoppedValue_apply] using h\n · have h := hZ.expected_stoppedValue_mono hP hn\n (fun ω => min_le_left _ _) (N := n) (fun ω => le_rfl)\n simpa only [stoppedValue_const, stoppedProcess_eq_stoppedValue_apply] using h", "time_seconds": 300, "topic": "Martingales & stopping" }, { "question_id": 96, "problem_name": "stochastic_processes_ctmc_reversibility_embedded", "informal_statement": "Let (X(t)) be an irreducible continuous-time Markov chain with embedded jump chain (X_n).\n\nShow that the process (X(t)) is reversible if and only if the embedded chain (X_n) is reversible.", "lean_statement": "theorem q96_ctmc_reversibility_embedded {S : Type*} [Countable S]\n (ν : S → ℝ) (hν : ∀ i, 0 < ν i)\n (P : S → S → ℝ) (hP0 : ∀ i j, 0 ≤ P i j) (hPloop : ∀ i, P i i = 0)\n (hPstoch : ∀ i, ∑' j, P i j = 1)\n (q : S → S → ℝ) (hq : ∀ i j, q i j = ν i * P i j)\n (p : S → ℝ) (hp0 : ∀ i, 0 < p i) (hpsum : ∑' i, p i = 1)\n (π : S → ℝ) (hπ0 : ∀ i, 0 < π i) (hπsum : ∑' i, π i = 1)\n (hnorm : Summable (fun k => p k * ν k))\n (hlink : ∀ i, π i * (∑' k, p k * ν k) = p i * ν i) :\n (∀ i j, p i * q i j = p j * q j i) ↔ (∀ i j, π i * P i j = π j * P j i) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q96_ctmc_reversibility_embedded {S : Type*} [Countable S]\n (ν : S → ℝ) (hν : ∀ i, 0 < ν i)\n (P : S → S → ℝ) (hP0 : ∀ i j, 0 ≤ P i j) (hPloop : ∀ i, P i i = 0)\n (hPstoch : ∀ i, ∑' j, P i j = 1)\n (q : S → S → ℝ) (hq : ∀ i j, q i j = ν i * P i j)\n (p : S → ℝ) (hp0 : ∀ i, 0 < p i) (hpsum : ∑' i, p i = 1)\n (π : S → ℝ) (hπ0 : ∀ i, 0 < π i) (hπsum : ∑' i, π i = 1)\n (hnorm : Summable (fun k => p k * ν k))\n (hlink : ∀ i, π i * (∑' k, p k * ν k) = p i * ν i) :\n (∀ i j, p i * q i j = p j * q j i) ↔ (∀ i j, π i * P i j = π j * P j i) := by\n have hkey : ∀ i j, π i * P i j * (∑' k, p k * ν k) = p i * q i j := by\n intro i j\n rw [hq i j]\n linear_combination (P i j) * hlink i\n have hne : Nonempty S := by\n rcases isEmpty_or_nonempty S with h | h\n · exfalso\n have hz : π = fun _ => (0:ℝ) := funext (fun i => (h.false i).elim)\n rw [hz, tsum_zero] at hπsum; norm_num at hπsum\n · exact h\n obtain ⟨i0⟩ := hne\n have hZ : (∑' k, p k * ν k) ≠ 0 := by\n intro h0\n have hli := hlink i0\n rw [h0, mul_zero] at hli\n have hpos : 0 < p i0 * ν i0 := mul_pos (hp0 i0) (hν i0)\n linarith [hli]\n constructor\n · intro hrev i j\n have e1 := hkey i j\n have e2 := hkey j i\n have hr := hrev i j\n have h3 : π i * P i j * (∑' k, p k * ν k) = π j * P j i * (∑' k, p k * ν k) := by\n rw [e1, e2, hr]\n exact mul_right_cancel₀ hZ h3\n · intro hrev i j\n have e1 := hkey i j\n have e2 := hkey j i\n have hr := hrev i j\n rw [← e1, ← e2, hr]", "time_seconds": 85, "topic": "Continuous-time Markov & queues" }, { "question_id": 99, "problem_name": "stochastic_processes_simple_random_walk", "informal_statement": "Let X = (X_n)_{n ≥ 0} be the simple random walk on ℤ.\n Let τ be the first time that the walker hits either 0 or N.\n\n Show that, for 0 ≤ k ≤ N, we have 𝔼_k [τ | X_τ = N] = (N^2 - k^2) / 3.", "lean_statement": "theorem q99_srw_conditional_time (N : ℕ) (v : ℕ → ℝ) (hbN : v N = 0)\n (hrec : ∀ k, 0 < k → k < N →\n v k = 1 + (((k : ℝ) + 1) / (2 * k)) * v (k + 1) + (((k : ℝ) - 1) / (2 * k)) * v (k - 1))\n (k : ℕ) (hk0 : 0 < k) (hk : k ≤ N) :\n v k = ((N : ℝ) ^ 2 - (k : ℝ) ^ 2) / 3 := by sorry", "true_to_source": "abstract", "proof": "theorem q99_srw_conditional_time (N : ℕ) (v : ℕ → ℝ) (hbN : v N = 0)\n (hrec : ∀ k, 0 < k → k < N →\n v k = 1 + (((k : ℝ) + 1) / (2 * k)) * v (k + 1) + (((k : ℝ) - 1) / (2 * k)) * v (k - 1))\n (k : ℕ) (hk0 : 0 < k) (hk : k ≤ N) :\n v k = ((N : ℝ) ^ 2 - (k : ℝ) ^ 2) / 3 := by\n -- Invariant: (j+1)*v(j+1) - j*v j = v 1 - (j+1)*j for j < N\n have I : ∀ j, j < N →\n ((j : ℝ) + 1) * v (j + 1) - (j : ℝ) * v j = v 1 - ((j : ℝ) + 1) * (j : ℝ) := by\n intro j\n induction j with\n | zero => intro _; simp\n | succ n ih =>\n intro hjN\n have hnN : n < N := Nat.lt_of_succ_lt hjN\n have ihn := ih hnN\n have hpos : 0 < n + 1 := Nat.succ_pos n\n have hrecn := hrec (n + 1) hpos hjN\n simp only [Nat.add_sub_cancel] at hrecn\n have hne : (2 * ((n : ℝ) + 1)) ≠ 0 := by positivity\n push_cast at hrecn ⊢\n have hrec2 : ((n : ℝ) + 1) * v (n + 1) =\n (2 * ((n : ℝ) + 1))\n + ((n : ℝ) + 1 + 1) * v (n + 1 + 1) + ((n : ℝ) + 1 - 1) * v n\n - ((n : ℝ) + 1) * v (n + 1) := by\n have := hrecn\n field_simp at this\n linarith [this]\n nlinarith [ihn, hrec2]\n -- Closed form: v (j+1) = v 1 - (((j+1)^2 - 1)/3) for j < N\n have C : ∀ j, j < N → v (j + 1) = v 1 - (((j : ℝ) + 1) ^ 2 - 1) / 3 := by\n intro j\n induction j with\n | zero => intro _; norm_num\n | succ n ih =>\n intro hjN\n have hnN : n < N := Nat.lt_of_succ_lt hjN\n have Cn := ih hnN\n have In := I (n + 1) hjN\n push_cast at In ⊢\n nlinarith [In, Cn]\n have hN1 : 1 ≤ N := by omega\n -- v N determines v 1\n have hCN := C (N - 1) (by omega)\n rw [Nat.sub_add_cancel hN1] at hCN\n have hcastN : ((N : ℝ) - 1) = ((N - 1 : ℕ) : ℝ) := by\n rw [Nat.cast_sub hN1]; norm_num\n have hv1 : v 1 = ((N : ℝ) ^ 2 - 1) / 3 := by\n have : (0 : ℝ) = v 1 - ((((N - 1 : ℕ) : ℝ) + 1) ^ 2 - 1) / 3 := by rw [← hbN]; exact hCN\n have hc : ((N - 1 : ℕ) : ℝ) + 1 = (N : ℝ) := by\n rw [Nat.cast_sub hN1]; norm_num\n rw [hc] at this\n linarith [this]\n -- conclude\n have hCk := C (k - 1) (by omega)\n rw [Nat.sub_add_cancel hk0] at hCk\n have hck : ((k - 1 : ℕ) : ℝ) + 1 = (k : ℝ) := by\n rw [Nat.cast_sub hk0]; norm_num\n rw [hck] at hCk\n rw [hCk, hv1]\n ring", "time_seconds": 675, "topic": "Random walks & large deviations" }, { "question_id": 101, "problem_name": "stochastic_processes_optional_stopping_bounded_process_martingale", "informal_statement": "Let (X_n) be a martingale that is uniformly bounded, i.e. |X_n| ≤ M almost surely for all n, and let τ be a stopping time.\n\nShow that E[X_τ] = E[X_0].", "lean_statement": "theorem q101_optional_stopping_bounded_process_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ)\n (hX : Martingale X ℱ μ) (M : ℝ) (hbdd : ∀ n ω, |X n ω| ≤ M)\n (T : Ω → ℕ) (hT : IsStoppingTime ℱ (natStop T)) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ = ∫ ω, X 0 ω ∂μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q101_optional_stopping_bounded_process_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ)\n (hX : Martingale X ℱ μ) (M : ℝ) (hbdd : ∀ n ω, |X n ω| ≤ M)\n (T : Ω → ℕ) (hT : IsStoppingTime ℱ (natStop T)) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ = ∫ ω, X 0 ω ∂μ := by\n have hYsub : Submartingale (stoppedProcess X (natStop T)) ℱ μ :=\n hX.submartingale.stoppedProcess hT\n have hnYsub : Submartingale (stoppedProcess (-X) (natStop T)) ℱ μ :=\n hX.neg.submartingale.stoppedProcess hT\n have hval : ∀ ω, stoppedValue X (natStop T) ω = X (T ω) ω := by\n intro ω; rfl\n have hzero : ∀ ω, stoppedProcess X (natStop T) 0 ω = X 0 ω := by\n intro ω\n rw [MeasureTheory.stoppedProcess_eq_of_le (by simp only [natStop]; exact_mod_cast Nat.zero_le (T ω))]\n have hEqXn : ∀ n, ∫ ω, stoppedProcess X (natStop T) n ω ∂μ = ∫ ω, X 0 ω ∂μ := by\n intro n\n have hle1 : ∫ ω, stoppedProcess X (natStop T) 0 ω ∂μ\n ≤ ∫ ω, stoppedProcess X (natStop T) n ω ∂μ := by\n have := hYsub.setIntegral_le (Nat.zero_le n) MeasurableSet.univ\n simpa only [setIntegral_univ] using this\n have hle2 : ∫ ω, stoppedProcess X (natStop T) n ω ∂μ\n ≤ ∫ ω, stoppedProcess X (natStop T) 0 ω ∂μ := by\n have hraw := hnYsub.setIntegral_le (Nat.zero_le n) MeasurableSet.univ\n simp only [setIntegral_univ] at hraw\n have e0 : ∫ ω, stoppedProcess (-X) (natStop T) 0 ω ∂μ\n = -∫ ω, stoppedProcess X (natStop T) 0 ω ∂μ := by\n rw [← integral_neg]; rfl\n have en : ∫ ω, stoppedProcess (-X) (natStop T) n ω ∂μ\n = -∫ ω, stoppedProcess X (natStop T) n ω ∂μ := by\n rw [← integral_neg]; rfl\n rw [e0, en] at hraw\n linarith\n have heq0 : ∫ ω, stoppedProcess X (natStop T) 0 ω ∂μ = ∫ ω, X 0 ω ∂μ :=\n integral_congr_ae (ae_of_all μ hzero)\n have hEq : ∫ ω, stoppedProcess X (natStop T) n ω ∂μ\n = ∫ ω, stoppedProcess X (natStop T) 0 ω ∂μ := le_antisymm hle2 hle1\n rw [hEq, heq0]\n have hmeas : ∀ n, AEStronglyMeasurable (fun ω => stoppedProcess X (natStop T) n ω) μ :=\n fun n => ((hYsub.stronglyMeasurable n).mono (ℱ.le n)).aestronglyMeasurable\n have hbound : ∀ n, ∀ᵐ ω ∂μ, ‖stoppedProcess X (natStop T) n ω‖ ≤ M :=\n fun n => ae_of_all μ fun ω => by rw [Real.norm_eq_abs]; exact hbdd _ ω\n have hlim : ∀ᵐ ω ∂μ, Tendsto (fun n => stoppedProcess X (natStop T) n ω) atTop\n (𝓝 (stoppedValue X (natStop T) ω)) :=\n ae_of_all μ fun ω => tendsto_atTop_of_eventually_const (i₀ := T ω) fun n hn => by\n have hle : natStop T ω ≤ (↑n : WithTop ℕ) := by simp only [natStop]; exact_mod_cast hn\n rw [MeasureTheory.stoppedProcess_eq_of_ge (u := X) (i := n) hle, hval ω]; rfl\n have hdct := tendsto_integral_of_dominated_convergence (fun _ => M) hmeas\n (integrable_const M) hbound hlim\n have hconst : Tendsto (fun n => ∫ ω, stoppedProcess X (natStop T) n ω ∂μ) atTop\n (𝓝 (∫ ω, X 0 ω ∂μ)) := by\n have hfun : (fun n => ∫ ω, stoppedProcess X (natStop T) n ω ∂μ)\n = fun _ => ∫ ω, X 0 ω ∂μ := funext hEqXn\n rw [hfun]; exact tendsto_const_nhds\n exact tendsto_nhds_unique hdct hconst", "time_seconds": 1110, "topic": "Martingales & stopping" }, { "question_id": 102, "problem_name": "stochastic_processes_chapman_kolmogorov_finite", "informal_statement": "Let (X_n) be a finite-state Markov chain with transition matrix [P].\n\nShow that the n-step transition probabilities satisfy P_{ij}^{m+n} = Σ_k P_{ik}^m P_{kj}^n.", "lean_statement": "theorem q102_chapman_kolmogorov {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (m n : ℕ) (i j : Ω) :\n (P ^ (m + n)) i j = ∑ k, (P ^ m) i k * (P ^ n) k j := by\n sorry", "true_to_source": "literal", "proof": "theorem q102_chapman_kolmogorov {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (m n : ℕ) (i j : Ω) :\n (P ^ (m + n)) i j = ∑ k, (P ^ m) i k * (P ^ n) k j := by\n rw [pow_add, Matrix.mul_apply]", "time_seconds": 47, "topic": "Markov chains (finite & countable)" }, { "question_id": 103, "problem_name": "stochastic_processes_random_walk_centered_martingale", "informal_statement": "Let X_1,X_2,… be i.i.d. with mean μ and E|X_1| < ∞, and let S_n = Σ_{i=1}^n X_i.\n\nShow that S_n − μn is a martingale with respect to the filtration generated by (X_i).", "lean_statement": "theorem q103_centered_random_walk_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (X : ℕ → Ω → ℝ)\n (hadap : Adapted ℱ (fun n ω => ∑ i ∈ Finset.range n, X i ω))\n (hint : ∀ n, Integrable (X n) μ) (m : ℝ)\n (hincr : ∀ n, μ[X n | ℱ n] =ᵐ[μ] fun _ => m) :\n Martingale (fun n ω => (∑ i ∈ Finset.range n, X i ω) - m * (n : ℝ)) ℱ μ := by\n sorry", "true_to_source": "abstract", "proof": "theorem q103_centered_random_walk_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (X : ℕ → Ω → ℝ)\n (hadap : Adapted ℱ (fun n ω => ∑ i ∈ Finset.range n, X i ω))\n (hint : ∀ n, Integrable (X n) μ) (m : ℝ)\n (hincr : ∀ n, μ[X n | ℱ n] =ᵐ[μ] fun _ => m) :\n Martingale (fun n ω => (∑ i ∈ Finset.range n, X i ω) - m * (n : ℝ)) ℱ μ := by\n refine martingale_nat ?_ ?_ ?_\n · intro n\n exact (hadap.stronglyAdapted n).sub stronglyMeasurable_const\n · intro n\n exact (integrable_finset_sum _ (fun i _ => hint i)).sub (integrable_const _)\n · intro n\n have hle : ℱ n ≤ m0 := ℱ.le n\n have hSint : Integrable (fun ω => ∑ i ∈ Finset.range n, X i ω) μ :=\n integrable_finset_sum _ (fun i _ => hint i)\n have hXint := hint n\n have hAint : Integrable (fun ω => (∑ i ∈ Finset.range n, X i ω) - m * (n:ℝ)) μ :=\n hSint.sub (integrable_const _)\n have hBint : Integrable (fun ω => X n ω - m) μ := hXint.sub (integrable_const _)\n have h1 :\n (fun ω => (∑ i ∈ Finset.range (n+1), X i ω) - m * ((n+1 : ℕ):ℝ))\n = (fun ω => ((∑ i ∈ Finset.range n, X i ω) - m * (n:ℝ)) + (X n ω - m)) := by\n funext ω; rw [Finset.sum_range_succ]; push_cast; ring\n have hce :\n μ[fun ω => (∑ i ∈ Finset.range (n+1), X i ω) - m * ((n+1 : ℕ):ℝ) | ℱ n]\n =ᵐ[μ] fun ω => (∑ i ∈ Finset.range n, X i ω) - m * (n:ℝ) := by\n rw [h1]\n have hadd := condExp_add hAint hBint (ℱ n)\n refine hadd.trans ?_\n have hA : μ[fun ω => (∑ i ∈ Finset.range n, X i ω) - m * (n:ℝ) | ℱ n]\n =ᵐ[μ] fun ω => (∑ i ∈ Finset.range n, X i ω) - m * (n:ℝ) :=\n (condExp_of_stronglyMeasurable hle\n ((hadap.stronglyAdapted n).sub stronglyMeasurable_const) hAint).eventuallyEq\n have hB : μ[fun ω => X n ω - m | ℱ n] =ᵐ[μ] fun _ => (0:ℝ) := by\n have hsub : μ[fun ω => X n ω - m | ℱ n]\n =ᵐ[μ] μ[X n|ℱ n] - μ[fun _ => m|ℱ n] := condExp_sub hXint (integrable_const _) _\n refine hsub.trans ?_\n have hcm : μ[fun _ => m|ℱ n] =ᵐ[μ] fun _ => m := (condExp_const hle m).eventuallyEq\n filter_upwards [hincr n, hcm] with ω h1' h2'\n simp only [Pi.sub_apply, h1', h2', sub_self]\n filter_upwards [hA, hB] with ω hA' hB'\n simp only [Pi.add_apply, hA', hB', add_zero]\n exact hce.symm", "time_seconds": 173, "topic": "Martingales & stopping" }, { "question_id": 106, "problem_name": "stochastic_processes_birth_death_process_reversible", "informal_statement": "Consider a birth-death continuous-time Markov chain that possesses a steady-state distribution.\n\nShow that the process is reversible.", "lean_statement": "theorem q106_birth_death_reversible\n (lam : ℕ → ℝ) (hlam : ∀ i, 0 < lam i)\n (mu : ℕ → ℝ) (hmu : ∀ i, 0 < mu (i + 1))\n (q : ℕ → ℕ → ℝ)\n (hqbirth : ∀ i, q i (i + 1) = lam i)\n (hqdeath : ∀ i, q (i + 1) i = mu (i + 1))\n (hqother : ∀ i j, j ≠ i + 1 → i ≠ j + 1 → q i j = 0)\n (p : ℕ → ℝ) (hp0 : ∀ i, 0 ≤ p i) (hpsum : ∑' i, p i = 1)\n (hbal0 : p 0 * lam 0 = p 1 * mu 1)\n (hbal : ∀ n, p (n + 1) * (lam (n + 1) + mu (n + 1)) = p n * lam n + p (n + 2) * mu (n + 2)) :\n ∀ i j, p i * q i j = p j * q j i := by\n sorry", "true_to_source": "abstract", "proof": "theorem q106_birth_death_reversible\n (lam : ℕ → ℝ) (hlam : ∀ i, 0 < lam i)\n (mu : ℕ → ℝ) (hmu : ∀ i, 0 < mu (i + 1))\n (q : ℕ → ℕ → ℝ)\n (hqbirth : ∀ i, q i (i + 1) = lam i)\n (hqdeath : ∀ i, q (i + 1) i = mu (i + 1))\n (hqother : ∀ i j, j ≠ i + 1 → i ≠ j + 1 → q i j = 0)\n (p : ℕ → ℝ) (hp0 : ∀ i, 0 ≤ p i) (hpsum : ∑' i, p i = 1)\n (hbal0 : p 0 * lam 0 = p 1 * mu 1)\n (hbal : ∀ n, p (n + 1) * (lam (n + 1) + mu (n + 1)) = p n * lam n + p (n + 2) * mu (n + 2)) :\n ∀ i j, p i * q i j = p j * q j i := by\n have db : ∀ n, p n * lam n = p (n + 1) * mu (n + 1) := by\n intro n\n induction n with\n | zero => exact hbal0\n | succ k ih =>\n have e : p (k + 1) * (lam (k + 1) + mu (k + 1))\n = p (k + 1) * lam (k + 1) + p (k + 1) * mu (k + 1) := by ring\n have h := hbal k\n rw [e] at h\n linarith\n intro i j\n rcases eq_or_ne j (i + 1) with hj | hj\n · subst hj\n rw [hqbirth i, hqdeath i]\n exact db i\n · rcases eq_or_ne i (j + 1) with hi | hi\n · subst hi\n rw [hqdeath j, hqbirth j]\n exact (db j).symm\n · rw [hqother i j hj hi, hqother j i hi hj]\n ring", "time_seconds": 88, "topic": "Continuous-time Markov & queues" }, { "question_id": 107, "problem_name": "stochastic_processes_product_unit_mean_martingale", "informal_statement": "Let (X_i) be i.i.d. with E[X_i] = 1, and let Z_n = X_1 X_2 … X_n.\n\nShow that (Z_n) is a martingale.", "lean_statement": "theorem q107_iid_product_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hmeas : ∀ i, StronglyMeasurable (X i))\n (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (hint : ∀ n, Integrable (fun ω => ∏ i ∈ Finset.range (n + 1), X i ω) μ)\n (hmean : ∫ ω, X 0 ω ∂μ = 1) :\n Martingale (fun n ω => ∏ i ∈ Finset.range (n + 1), X i ω)\n (Filtration.natural X hmeas) μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q107_iid_product_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hmeas : ∀ i, StronglyMeasurable (X i))\n (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (hint : ∀ n, Integrable (fun ω => ∏ i ∈ Finset.range (n + 1), X i ω) μ)\n (hmean : ∫ ω, X 0 ω ∂μ = 1) :\n Martingale (fun n ω => ∏ i ∈ Finset.range (n + 1), X i ω)\n (Filtration.natural X hmeas) μ := by\n set ℱ := Filtration.natural X hmeas with hℱ\n have hadapt : StronglyAdapted ℱ X := Filtration.stronglyAdapted_natural hmeas\n have hXmeas : ∀ i k, k ≤ i → StronglyMeasurable[ℱ i] (X k) := by\n intro i k hk\n exact (hadapt k).mono (ℱ.mono hk)\n have hprodfun : ∀ m : ℕ, (fun ω => ∏ k ∈ Finset.range (m + 1), X k ω)\n = ∏ k ∈ Finset.range (m + 1), X k := fun m => by funext ω; rw [Finset.prod_apply]\n have hAsm : ∀ i, StronglyMeasurable[ℱ i] (fun ω => ∏ k ∈ Finset.range (i + 1), X k ω) := by\n intro i\n rw [hprodfun i]\n exact Finset.stronglyMeasurable_prod _\n (fun k hk => hXmeas i k (Nat.lt_succ_iff.mp (Finset.mem_range.mp hk)))\n have hX0int : Integrable (X 0) μ := by\n by_contra h; rw [integral_undef h] at hmean; exact zero_ne_one hmean\n have hXint1 : ∀ k, ∫ ω, X k ω ∂μ = 1 := fun k => (hident k).integral_eq.trans hmean\n have hXkint : ∀ k, Integrable (X k) μ := fun k => (hident k).integrable_iff.mpr hX0int\n refine martingale_nat hAsm hint (fun i => ?_)\n -- one-step: f i =ᵐ μ[f (i+1) | ℱ i]\n have hg : Integrable (X (i + 1)) μ := hXkint (i + 1)\n have hfg : Integrable ((fun ω => ∏ k ∈ Finset.range (i + 1), X k ω) * X (i + 1)) μ := by\n have he : ((fun ω => ∏ k ∈ Finset.range (i + 1), X k ω) * X (i + 1))\n = (fun ω => ∏ k ∈ Finset.range (i + 1 + 1), X k ω) := by\n funext ω; rw [Pi.mul_apply]; exact (Finset.prod_range_succ (fun k => X k ω) (i + 1)).symm\n rw [he]; exact hint (i + 1)\n have hcond : μ[X (i + 1) | (ℱ i : MeasurableSpace Ω)] =ᵐ[μ] fun _ => ∫ ω, X (i + 1) ω ∂μ :=\n hindep.condExp_natural_ae_eq_of_lt hmeas (Nat.lt_succ_self i)\n have hstep : μ[fun ω => ∏ k ∈ Finset.range (i + 1 + 1), X k ω | (ℱ i : MeasurableSpace Ω)]\n =ᵐ[μ] fun ω => ∏ k ∈ Finset.range (i + 1), X k ω := by\n have he : (fun ω => ∏ k ∈ Finset.range (i + 1 + 1), X k ω)\n = (fun ω => ∏ k ∈ Finset.range (i + 1), X k ω) * X (i + 1) := by\n funext ω; rw [Pi.mul_apply, Finset.prod_range_succ]\n rw [he]\n calc μ[(fun ω => ∏ k ∈ Finset.range (i + 1), X k ω) * X (i + 1) | (ℱ i : MeasurableSpace Ω)]\n =ᵐ[μ] (fun ω => ∏ k ∈ Finset.range (i + 1), X k ω)\n * μ[X (i + 1) | (ℱ i : MeasurableSpace Ω)] :=\n condExp_mul_of_aestronglyMeasurable_left (hAsm i).aestronglyMeasurable hfg hg\n _ =ᵐ[μ] fun ω => ∏ k ∈ Finset.range (i + 1), X k ω := by\n filter_upwards [hcond] with ω hω\n simp only [Pi.mul_apply, hω, hXint1 (i + 1), mul_one]\n exact hstep.symm", "time_seconds": 265, "topic": "Martingales & stopping" }, { "question_id": 110, "problem_name": "stochastic_processes_brownian_scaling_1", "informal_statement": "Standard Brownian motion is invariant under Brownian scaling: for c > 0, the rescaled process W(t) = c B(t / c²) has the same increment law as B, namely W(t) − W(s) ~ Normal(0, t − s) for 0 ≤ s ≤ t.", "lean_statement": "theorem q110_brownian_scaling (μ : Measure Ω) [IsProbabilityMeasure μ]\n (B : ℝ → Ω → ℝ) (c : ℝ) (hc : 0 < c)\n (hincr : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n HasLaw (fun ω => B t ω - B s ω) (gaussianReal 0 (t - s).toNNReal) μ) :\n ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n HasLaw (fun ω => c * B (t / c ^ 2) ω - c * B (s / c ^ 2) ω)\n (gaussianReal 0 (t - s).toNNReal) μ := by\n sorry", "true_to_source": "abstract", "proof": "theorem q110_brownian_scaling (μ : Measure Ω) [IsProbabilityMeasure μ]\n (B : ℝ → Ω → ℝ) (c : ℝ) (hc : 0 < c)\n (hincr : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n HasLaw (fun ω => B t ω - B s ω) (gaussianReal 0 (t - s).toNNReal) μ) :\n ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n HasLaw (fun ω => c * B (t / c ^ 2) ω - c * B (s / c ^ 2) ω)\n (gaussianReal 0 (t - s).toNNReal) μ := by\n intro s t hs hst\n have hcne : (c:ℝ) ≠ 0 := ne_of_gt hc\n have hc2 : (0:ℝ) < c ^ 2 := by positivity\n have hs' : 0 ≤ s / c ^ 2 := by positivity\n have hst' : s / c ^ 2 ≤ t / c ^ 2 := by gcongr\n have hY := hincr (s / c ^ 2) (t / c ^ 2) hs' hst'\n have hmul := gaussianReal_const_mul hY c\n have h1 : (0:ℝ) ≤ t / c ^ 2 - s / c ^ 2 := by rw [sub_nonneg]; exact hst'\n have h2 : (0:ℝ) ≤ t - s := by linarith\n have hfun : (fun ω => c * B (t / c ^ 2) ω - c * B (s / c ^ 2) ω)\n = (fun ω => c * (B (t / c ^ 2) ω - B (s / c ^ 2) ω)) := by\n funext ω; ring\n rw [hfun]\n convert hmul using 2\n · exact (mul_zero c).symm\n · apply NNReal.coe_injective\n rw [NNReal.coe_mul, NNReal.coe_mk, Real.coe_toNNReal _ h1, Real.coe_toNNReal _ h2]\n field_simp", "time_seconds": 303, "topic": "Brownian motion & stochastic calculus" }, { "question_id": 113, "problem_name": "stochastic_processes_gamblers_ruin_1", "informal_statement": "Consider a gambler betting on the outcome of a sequence of independent fair coin tosses.\n If head, he gains one dollar. If tail, he loses one dollar.\n If he reaches a fortune of N dollars, he stops. If his purse is ever empty, he stops.\n \n Show that the probability that the gambler's fortune is equal to N dollars when they stop is k/N.", "lean_statement": "theorem q113_gamblers_ruin_probability\n (N : ℕ) (hN : 0 < N) (h : ℕ → ℝ)\n (hb0 : h 0 = 0) (hbN : h N = 1)\n (hharm : ∀ k, 0 < k → k < N → h k = (h (k - 1) + h (k + 1)) / 2)\n (k : ℕ) (hk : k ≤ N) :\n h k = (k : ℝ) / (N : ℝ) := by sorry", "true_to_source": "literal", "proof": "theorem q113_gamblers_ruin_probability\n (N : ℕ) (hN : 0 < N) (h : ℕ → ℝ)\n (hb0 : h 0 = 0) (hbN : h N = 1)\n (hharm : ∀ k, 0 < k → k < N → h k = (h (k - 1) + h (k + 1)) / 2)\n (k : ℕ) (hk : k ≤ N) :\n h k = (k : ℝ) / (N : ℝ) := by\n -- Express everything in terms of the slope h 1.\n have key : ∀ j, j ≤ N → h j = (j : ℝ) * h 1 := by\n intro j\n induction j using Nat.strong_induction_on with\n | _ j ih =>\n intro hj\n match j with\n | 0 => simpa using hb0\n | 1 => simp\n | (m + 2) =>\n have hk1 : 0 < m + 1 := Nat.succ_pos m\n have hk2 : m + 1 < N := by omega\n have hrec := hharm (m + 1) hk1 hk2\n have e1 : h m = (m : ℝ) * h 1 := ih m (by omega) (by omega)\n have e2 : h (m + 1) = ((m : ℝ) + 1) * h 1 := by\n have := ih (m + 1) (by omega) (by omega); push_cast at this; linarith\n have hstep : h (m + 1 + 1) = 2 * h (m + 1) - h m := by\n have hh : h (m + 1) = (h m + h (m + 1 + 1)) / 2 := by\n simpa using hrec\n linarith\n rw [hstep, e1, e2]; push_cast; ring\n have hNpos : (N : ℝ) ≠ 0 := by exact_mod_cast hN.ne'\n have hslope : h 1 = 1 / (N : ℝ) := by\n have hkN := key N (le_refl N)\n rw [hbN] at hkN\n field_simp at hkN ⊢\n linarith [hkN]\n rw [key k hk, hslope]; ring", "time_seconds": 80, "topic": "Random walks & large deviations" }, { "question_id": 119, "problem_name": "stochastic_processes_log_likelihood_random_walk", "informal_statement": "Consider a binary hypothesis test with i.i.d. observations Y_1, Y_2, … and let Z_n be the log-likelihood ratio after n observations.\n\nShow that under each hypothesis (Z_n) is a random walk whose increments are the per-observation log-likelihood ratios.", "lean_statement": "theorem q119_log_likelihood_random_walk (μ : Measure Ω) [IsProbabilityMeasure μ]\n (Y : ℕ → Ω → ℝ) (hindep : iIndepFun Y μ) (hident : ∀ i, IdentDistrib (Y i) (Y 0) μ μ)\n (llr : ℝ → ℝ) (hllr : Measurable llr)\n (W : ℕ → Ω → ℝ) (hW : ∀ i ω, W i ω = llr (Y i ω))\n (Z : ℕ → Ω → ℝ) (hZ : ∀ n ω, Z n ω = ∑ i ∈ Finset.range n, W i ω) :\n iIndepFun W μ ∧ (∀ i, IdentDistrib (W i) (W 0) μ μ) ∧\n (∀ n ω, Z (n + 1) ω - Z n ω = W n ω) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q119_log_likelihood_random_walk (μ : Measure Ω) [IsProbabilityMeasure μ]\n (Y : ℕ → Ω → ℝ) (hindep : iIndepFun Y μ) (hident : ∀ i, IdentDistrib (Y i) (Y 0) μ μ)\n (llr : ℝ → ℝ) (hllr : Measurable llr)\n (W : ℕ → Ω → ℝ) (hW : ∀ i ω, W i ω = llr (Y i ω))\n (Z : ℕ → Ω → ℝ) (hZ : ∀ n ω, Z n ω = ∑ i ∈ Finset.range n, W i ω) :\n iIndepFun W μ ∧ (∀ i, IdentDistrib (W i) (W 0) μ μ) ∧\n (∀ n ω, Z (n + 1) ω - Z n ω = W n ω) := by\n have hWeq : W = fun i => llr ∘ (Y i) := by\n funext i ω; simp [hW, Function.comp]\n refine ⟨?_, ?_, ?_⟩\n · rw [hWeq]\n exact hindep.comp (fun _ => llr) (fun _ => hllr)\n · intro i\n rw [hWeq]\n exact (hident i).comp hllr\n · intro n ω\n rw [hZ, hZ, Finset.sum_range_succ]\n ring", "time_seconds": 90, "topic": "Random walks & large deviations" }, { "question_id": 121, "problem_name": "stochastic_processes_martingale_stopping_time_3", "informal_statement": "Let (X_n)_{n ≥ 0} be a martingale, and let T be a stopping time which is finite a.s.\n Let there be an integrable random variable Y such that |Xn| ≤ Y for all n.\n\n Show that 𝔼[X_T] = 𝔼[X_0].", "lean_statement": "theorem q121_optional_stopping_dominated (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Martingale X ℱ μ) (T : Ω → ℕ)\n (hT : IsStoppingTime ℱ (natStop T)) (Y : Ω → ℝ) (hY : Integrable Y μ)\n (hdom : ∀ n, ∀ᵐ ω ∂μ, |X n ω| ≤ Y ω) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ = ∫ ω, X 0 ω ∂μ := by sorry", "true_to_source": "literal", "proof": "theorem q121_optional_stopping_dominated (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Martingale X ℱ μ) (T : Ω → ℕ)\n (hT : IsStoppingTime ℱ (natStop T)) (Y : Ω → ℝ) (hY : Integrable Y μ)\n (hdom : ∀ n, ∀ᵐ ω ∂μ, |X n ω| ≤ Y ω) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ = ∫ ω, X 0 ω ∂μ := by\n -- stopped process is a martingale\n have hSP : Martingale (stoppedProcess X (natStop T)) ℱ μ := by\n rw [martingale_iff]\n refine ⟨?_, hX.submartingale.stoppedProcess hT⟩\n have hsub : Submartingale (stoppedProcess (-X) (natStop T)) ℱ μ :=\n hX.neg.submartingale.stoppedProcess hT\n have hkey : -(stoppedProcess (-X) (natStop T)) = stoppedProcess X (natStop T) := by\n funext i ω; simp [stoppedProcess, Pi.neg_apply]\n have := hsub.neg\n rwa [hkey] at this\n set F : ℕ → Ω → ℝ := fun N => stoppedProcess X (natStop T) N with hF\n -- constant expectation\n have hconst : ∀ N, ∫ ω, F N ω ∂μ = ∫ ω, X 0 ω ∂μ := by\n intro N\n have h := hSP.setIntegral_eq (Nat.zero_le N) (MeasurableSet.univ)\n simp only [setIntegral_univ] at h\n rw [hF]\n rw [← h]\n apply integral_congr_ae\n filter_upwards with ω\n exact stoppedProcess_eq_of_le (by unfold natStop; exact_mod_cast Nat.zero_le (T ω))\n -- dominated convergence limit\n have hlim : Tendsto (fun N => ∫ ω, F N ω ∂μ) atTop\n (𝓝 (∫ ω, stoppedValue X (natStop T) ω ∂μ)) := by\n apply tendsto_integral_of_dominated_convergence Y\n · intro N; exact ((hSP.stronglyMeasurable N).mono (ℱ.le N)).aestronglyMeasurable\n · exact hY\n · intro N\n filter_upwards [ae_all_iff.mpr hdom] with ω hω\n rw [Real.norm_eq_abs]\n exact hω _\n · filter_upwards with ω\n apply tendsto_atTop_of_eventually_const (i₀ := T ω)\n intro N hN\n rw [hF]\n exact stoppedProcess_eq_of_ge (by unfold natStop; exact WithTop.coe_le_coe.mpr hN)\n rw [tendsto_congr hconst] at hlim\n exact tendsto_nhds_unique hlim tendsto_const_nhds", "time_seconds": 304, "topic": "Martingales & stopping" }, { "question_id": 123, "problem_name": "stochastic_processes_stopped_supermartingale", "informal_statement": "Let (X_n) be a supermartingale and let τ be a stopping time.\n\nShow that the stopped process X_{n∧τ} is a supermartingale.", "lean_statement": "theorem q123_stopped_supermartingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ)\n (hX : Supermartingale X ℱ μ) (T : Ω → ℕ) (hT : IsStoppingTime ℱ (natStop T)) :\n Supermartingale (stoppedProcess X (natStop T)) ℱ μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q123_stopped_supermartingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ)\n (hX : Supermartingale X ℱ μ) (T : Ω → ℕ) (hT : IsStoppingTime ℱ (natStop T)) :\n Supermartingale (stoppedProcess X (natStop T)) ℱ μ := by\n have hsub : Submartingale (stoppedProcess (-X) (natStop T)) ℱ μ :=\n hX.neg.stoppedProcess hT\n have hkey : -(stoppedProcess (-X) (natStop T)) = stoppedProcess X (natStop T) := by\n funext i ω\n simp [stoppedProcess, Pi.neg_apply]\n have := hsub.neg\n rwa [hkey] at this", "time_seconds": 120, "topic": "Martingales & stopping" }, { "question_id": 124, "problem_name": "stochastic_processes_markov_chain_2", "informal_statement": "A company issues N different types of coupons. Each coupon is equally likely to be each of the N types.\n A collector desires a complete set. Let τ be the first time that the collector obtains all N types.\n\n Show that for any c > 0, P[τ > N log N + cN] ≤ e^{-c}.", "lean_statement": "theorem q124_coupon_collector_tail {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n [IsProbabilityMeasure μ] (N : ℕ) (hN : 0 < N) (c : ℝ) (hc : 0 < c)\n (X : ℕ → Ω → ℕ)\n (hXrange : ∀ k ω, X k ω < N)\n (hXunif : ∀ (k : ℕ) (j : ℕ), j < N → (μ {ω | X k ω = j}).toReal = 1 / (N : ℝ))\n (hXindep : iIndepFun X μ)\n (τ : Ω → ℕ)\n (hτ : ∀ ω, τ ω = sInf {m : ℕ | ∀ j, j < N → ∃ k, k < m ∧ X k ω = j}) :\n (μ {ω | ⌈(N : ℝ) * Real.log N + c * N⌉₊ < τ ω}).toReal ≤ Real.exp (-c) := by sorry", "true_to_source": "abstract", "proof": "theorem q124_coupon_collector_tail {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n [IsProbabilityMeasure μ] (N : ℕ) (hN : 0 < N) (c : ℝ) (hc : 0 < c)\n (X : ℕ → Ω → ℕ)\n (hXrange : ∀ k ω, X k ω < N)\n (hXunif : ∀ (k : ℕ) (j : ℕ), j < N → (μ {ω | X k ω = j}).toReal = 1 / (N : ℝ))\n (hXindep : iIndepFun X μ)\n (τ : Ω → ℕ)\n (hτ : ∀ ω, τ ω = sInf {m : ℕ | ∀ j, j < N → ∃ k, k < m ∧ X k ω = j}) :\n (μ {ω | ⌈(N : ℝ) * Real.log N + c * N⌉₊ < τ ω}).toReal ≤ Real.exp (-c) := by\n have hNR : (0:ℝ) < N := by exact_mod_cast hN\n have hNR0 : (N:ℝ) ≠ 0 := ne_of_gt hNR\n set m : ℕ := ⌈(N : ℝ) * Real.log N + c * N⌉₊ with hm\n rw [← MeasureTheory.measureReal_def]\n -- Step A: subset relation\n have hsub : {ω | m < τ ω} ⊆ ⋃ j ∈ Finset.range N, {ω | ∀ k < m, X k ω ≠ j} := by\n intro ω hω\n simp only [Set.mem_setOf_eq] at hω\n rw [hτ ω] at hω\n have hnot : ¬ (∀ j, j < N → ∃ k, k < m ∧ X k ω = j) := by\n intro hmem\n have hmemS : m ∈ {m' : ℕ | ∀ j, j < N → ∃ k, k < m' ∧ X k ω = j} := hmem\n exact absurd (Nat.sInf_le hmemS) (not_le.mpr hω)\n push_neg at hnot\n obtain ⟨j, hjN, hjk⟩ := hnot\n simp only [Set.mem_iUnion, Set.mem_setOf_eq, Finset.mem_range]\n exact ⟨j, hjN, hjk⟩\n -- Step B: union bound (in reals)\n have hbound1 : μ.real {ω | m < τ ω}\n ≤ ∑ j ∈ Finset.range N, μ.real {ω | ∀ k < m, X k ω ≠ j} := by\n calc μ.real {ω | m < τ ω}\n ≤ μ.real (⋃ j ∈ Finset.range N, {ω | ∀ k < m, X k ω ≠ j}) :=\n measureReal_mono hsub (measure_ne_top μ _)\n _ ≤ ∑ j ∈ Finset.range N, μ.real {ω | ∀ k < m, X k ω ≠ j} :=\n measureReal_biUnion_finset_le _ _\n -- Step C: per-type bound\n have key : ∀ j, j < N → μ.real {ω | ∀ k < m, X k ω ≠ j} ≤ (1 - 1/(N:ℝ))^m := by\n intro j hjN\n have hset : {ω | ∀ k < m, X k ω ≠ j} = ⋂ k ∈ Finset.range m, X k ⁻¹' ({j}ᶜ) := by\n ext ω\n simp only [Set.mem_setOf_eq, Set.mem_iInter, Set.mem_preimage, Set.mem_compl_iff,\n Set.mem_singleton_iff, Finset.mem_range]\n rw [hset]\n have hprod : μ (⋂ k ∈ Finset.range m, X k ⁻¹' ({j}ᶜ))\n = ∏ k ∈ Finset.range m, μ (X k ⁻¹' ({j}ᶜ)) := by\n exact (iIndepFun_iff_measure_inter_preimage_eq_mul).mp hXindep (Finset.range m)\n (sets := fun _ => {j}ᶜ) (fun i _ => (measurableSet_singleton j).compl)\n have hprodreal : μ.real (⋂ k ∈ Finset.range m, X k ⁻¹' ({j}ᶜ))\n = ∏ k ∈ Finset.range m, μ.real (X k ⁻¹' ({j}ᶜ)) := by\n simp only [MeasureTheory.measureReal_def]\n rw [hprod, ENNReal.toReal_prod]\n rw [hprodreal]\n have hfac : ∀ k, μ.real (X k ⁻¹' ({j}ᶜ)) ≤ 1 - 1/(N:ℝ) := by\n intro k\n have hsub2 : X k ⁻¹' ({j}ᶜ) ⊆ ⋃ i ∈ (Finset.range N).erase j, {ω | X k ω = i} := by\n intro ω hω\n simp only [Set.mem_preimage, Set.mem_compl_iff, Set.mem_singleton_iff] at hω\n simp only [Set.mem_iUnion, Set.mem_setOf_eq, Finset.mem_erase, Finset.mem_range]\n exact ⟨X k ω, ⟨hω, hXrange k ω⟩, rfl⟩\n calc μ.real (X k ⁻¹' ({j}ᶜ))\n ≤ μ.real (⋃ i ∈ (Finset.range N).erase j, {ω | X k ω = i}) :=\n measureReal_mono hsub2 (measure_ne_top μ _)\n _ ≤ ∑ i ∈ (Finset.range N).erase j, μ.real {ω | X k ω = i} :=\n measureReal_biUnion_finset_le _ _\n _ = ∑ i ∈ (Finset.range N).erase j, (1/(N:ℝ)) := by\n refine Finset.sum_congr rfl (fun i hi => ?_)\n rw [Finset.mem_erase, Finset.mem_range] at hi\n rw [MeasureTheory.measureReal_def]\n exact hXunif k i hi.2\n _ = 1 - 1/(N:ℝ) := by\n rw [Finset.sum_const, Finset.card_erase_of_mem (Finset.mem_range.mpr hjN),\n Finset.card_range, nsmul_eq_mul, Nat.cast_sub hN, Nat.cast_one, mul_one_div,\n sub_div, div_self hNR0]\n calc ∏ k ∈ Finset.range m, μ.real (X k ⁻¹' ({j}ᶜ))\n ≤ ∏ k ∈ Finset.range m, (1 - 1/(N:ℝ)) :=\n Finset.prod_le_prod (fun k _ => MeasureTheory.measureReal_nonneg) (fun k _ => hfac k)\n _ = (1 - 1/(N:ℝ))^m := by rw [Finset.prod_const, Finset.card_range]\n -- Step D: real algebra\n have hp_nonneg : (0:ℝ) ≤ 1 - 1/N := by\n rw [sub_nonneg, div_le_one hNR]; exact_mod_cast hN\n have hp_le : 1 - 1/(N:ℝ) ≤ Real.exp (-(1/N)) := by\n have := Real.add_one_le_exp (-(1/(N:ℝ))); linarith\n have hpow : (1 - 1/(N:ℝ))^m ≤ Real.exp (-((m:ℝ)/N)) := by\n calc (1 - 1/(N:ℝ))^m ≤ (Real.exp (-(1/N)))^m := by\n apply pow_le_pow_left₀ hp_nonneg hp_le\n _ = Real.exp ((m:ℝ) * -(1/N)) := (Real.exp_nat_mul (-(1/N)) m).symm\n _ = Real.exp (-((m:ℝ)/N)) := by congr 1; ring\n have hmge : (N:ℝ) * Real.log N + c * N ≤ (m:ℝ) := by\n rw [hm]; exact Nat.le_ceil _\n have hexpbound : (N:ℝ) * Real.exp (-((m:ℝ)/N)) ≤ Real.exp (-c) := by\n have h1 : -((m:ℝ)/N) ≤ -(Real.log N + c) := by\n have hle : Real.log N + c ≤ (m:ℝ)/N := by\n rw [le_div_iff₀ hNR]; nlinarith [hmge]\n linarith\n have h2 : Real.exp (-((m:ℝ)/N)) ≤ Real.exp (-(Real.log N + c)) := Real.exp_le_exp.mpr h1\n have h3 : Real.exp (-(Real.log N + c)) = (N:ℝ)⁻¹ * Real.exp (-c) := by\n rw [neg_add, Real.exp_add, Real.exp_neg, Real.exp_log hNR]\n calc (N:ℝ) * Real.exp (-((m:ℝ)/N))\n ≤ (N:ℝ) * Real.exp (-(Real.log N + c)) := by\n apply mul_le_mul_of_nonneg_left h2 (le_of_lt hNR)\n _ = (N:ℝ) * ((N:ℝ)⁻¹ * Real.exp (-c)) := by rw [h3]\n _ = Real.exp (-c) := by field_simp\n -- assemble\n calc μ.real {ω | m < τ ω}\n ≤ ∑ j ∈ Finset.range N, μ.real {ω | ∀ k < m, X k ω ≠ j} := hbound1\n _ ≤ ∑ j ∈ Finset.range N, (1 - 1/(N:ℝ))^m :=\n Finset.sum_le_sum (fun j hj => key j (Finset.mem_range.mp hj))\n _ = (N:ℝ) * (1 - 1/(N:ℝ))^m := by\n rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul]\n _ ≤ (N:ℝ) * Real.exp (-((m:ℝ)/N)) := by\n apply mul_le_mul_of_nonneg_left hpow (le_of_lt hNR)\n _ ≤ Real.exp (-c) := hexpbound\n", "time_seconds": 875, "topic": "Markov chains (finite & countable)" }, { "question_id": 126, "problem_name": "stochastic_processes_target_time", "informal_statement": "Suppose that (X_n)_{n ≥ 0} is an irreducible Markov chain with transition\n matrix P and stationary measure π.\n Let τ_x be the hitting time: τ_x = min{n ≥ 0 : X_n = x}.\n\n Show that the quantity sum_x (𝔼_a [τ_x]π(x)) does not depend on a.", "lean_statement": "theorem q126_target_time {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (hirr : IsIrreducible P)\n (g : Ω → Ω → ℝ) (hg : IsHittingSolution P g)\n (π : Ω → ℝ) (hπ : IsStationary P π) :\n ∀ a b : Ω, ∑ x, g a x * π x = ∑ x, g b x * π x := by sorry", "true_to_source": "abstract", "proof": "theorem q126_target_time {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (hirr : IsIrreducible P)\n (g : Ω → Ω → ℝ) (hg : IsHittingSolution P g)\n (π : Ω → ℝ) (hπ : IsStationary P π) :\n ∀ a b : Ω, ∑ x, g a x * π x = ∑ x, g b x * π x := by\n obtain ⟨hPnn, hProw⟩ := hP\n obtain ⟨hπnn, hπsum, hπstat⟩ := hπ\n set f : Ω → ℝ := fun a => ∑ x, g a x * π x with hf\n -- Core first-step identity, valid for all s, x.\n have core : ∀ s x : Ω,\n g s x - ∑ z, P s z * g z x\n = 1 + (if s = x then -(1 + ∑ z, P s z * g z x) else 0) := by\n intro s x\n rcases eq_or_ne s x with rfl | h\n · rw [hg s s, if_pos rfl, if_pos rfl]; ring\n · rw [hg s x, if_neg h, if_neg h]; ring\n -- Kac's formula: π x · 𝔼_x[τ_x^+] = 1.\n have hkac : ∀ x : Ω, π x * (1 + ∑ z, P x z * g z x) = 1 := by\n intro x\n have hswap : ∑ y, π y * ∑ z, P y z * g z x = ∑ z, π z * g z x := by\n simp_rw [Finset.mul_sum]\n rw [Finset.sum_comm]\n apply Finset.sum_congr rfl\n intro z _\n rw [← hπstat z, Finset.sum_mul]\n apply Finset.sum_congr rfl\n intro y _\n ring\n have hzero : ∑ y, π y * (g y x - ∑ z, P y z * g z x) = 0 := by\n simp_rw [mul_sub]\n rw [Finset.sum_sub_distrib, hswap]\n exact sub_self _\n have key : (0 : ℝ) = 1 - π x * (1 + ∑ z, P x z * g z x) := by\n rw [← hzero]\n have hcongr : ∀ y : Ω, π y * (g y x - ∑ z, P y z * g z x)\n = π y + (if y = x then π y * -(1 + ∑ z, P y z * g z x) else 0) := by\n intro y\n rw [core y x, mul_add, mul_one, mul_ite, mul_zero]\n rw [Finset.sum_congr rfl (fun y _ => hcongr y), Finset.sum_add_distrib, hπsum,\n Finset.sum_ite_eq', if_pos (Finset.mem_univ x)]\n ring\n linarith [key]\n -- f is harmonic: P f = f.\n have hharm : ∀ a : Ω, ∑ z, P a z * f z = f a := by\n intro a\n have hswap2 : ∑ z, P a z * f z = ∑ x, π x * ∑ z, P a z * g z x := by\n simp only [hf]\n simp_rw [Finset.mul_sum]\n rw [Finset.sum_comm]\n apply Finset.sum_congr rfl\n intro x _\n apply Finset.sum_congr rfl\n intro z _\n ring\n have hfa : f a = ∑ x, π x * g a x := by\n simp only [hf]; apply Finset.sum_congr rfl; intro x _; ring\n have hdiff : ∑ x, π x * g a x - ∑ x, π x * ∑ z, P a z * g z x = 0 := by\n rw [← Finset.sum_sub_distrib]\n have hcongr : ∀ x : Ω, π x * g a x - π x * ∑ z, P a z * g z x\n = π x + (if a = x then π x * -(1 + ∑ z, P a z * g z x) else 0) := by\n intro x\n rw [← mul_sub, core a x, mul_add, mul_one, mul_ite, mul_zero]\n rw [Finset.sum_congr rfl (fun x _ => hcongr x), Finset.sum_add_distrib, hπsum,\n Finset.sum_ite_eq, if_pos (Finset.mem_univ a), mul_neg]\n linarith [hkac a]\n rw [hswap2, hfa]\n linarith [hdiff]\n -- Powers of a stochastic matrix are stochastic.\n have hPn_nn : ∀ (n : ℕ) (x y : Ω), 0 ≤ (P ^ n) x y := by\n intro n\n induction n with\n | zero => intro x y; rw [pow_zero, Matrix.one_apply]; split_ifs <;> norm_num\n | succ k ih =>\n intro x y\n rw [pow_succ, Matrix.mul_apply]\n exact Finset.sum_nonneg (fun z _ => mul_nonneg (ih x z) (hPnn z y))\n have hPn_row : ∀ (n : ℕ) (x : Ω), ∑ y, (P ^ n) x y = 1 := by\n intro n\n induction n with\n | zero => intro x; simp [pow_zero, Matrix.one_apply]\n | succ k ih =>\n intro x\n rw [pow_succ]\n simp_rw [Matrix.mul_apply]\n rw [Finset.sum_comm]\n simp_rw [← Finset.mul_sum, hProw, mul_one]\n exact ih x\n -- Harmonicity lifts to all powers.\n have hPnf : ∀ (n : ℕ) (x : Ω), ∑ y, (P ^ n) x y * f y = f x := by\n intro n\n induction n with\n | zero => intro x; simp [pow_zero, Matrix.one_apply]\n | succ k ih =>\n intro x\n rw [pow_succ]\n simp_rw [Matrix.mul_apply, Finset.sum_mul]\n rw [Finset.sum_comm]\n have hstep : ∀ z, (∑ y, (P ^ k) x z * P z y * f y) = (P ^ k) x z * f z := by\n intro z\n have hrw : (∑ y, (P ^ k) x z * P z y * f y) = (P ^ k) x z * ∑ y, P z y * f y := by\n rw [Finset.mul_sum]\n apply Finset.sum_congr rfl\n intro y _\n ring\n rw [hrw, hharm z]\n rw [Finset.sum_congr rfl (fun z _ => hstep z)]\n exact ih x\n -- Maximum principle: a harmonic function on an irreducible chain is constant.\n intro a b\n haveI : Nonempty Ω := ⟨a⟩\n obtain ⟨x0, hx0⟩ := Finite.exists_max f\n have hconst : ∀ y, f y = f x0 := by\n intro y\n obtain ⟨n, _, hnpos⟩ := hirr x0 y\n have hsum0 : ∑ z, (P ^ n) x0 z * (f x0 - f z) = 0 := by\n simp_rw [mul_sub]\n rw [Finset.sum_sub_distrib, ← Finset.sum_mul, hPn_row n x0, one_mul, hPnf n x0, sub_self]\n have hnn : ∀ z ∈ Finset.univ, 0 ≤ (P ^ n) x0 z * (f x0 - f z) := by\n intro z _\n exact mul_nonneg (hPn_nn n x0 z) (by linarith [hx0 z])\n have hterms := (Finset.sum_eq_zero_iff_of_nonneg hnn).mp hsum0\n have hy := hterms y (Finset.mem_univ y)\n have hfy : f x0 - f y = 0 := by\n rcases mul_eq_zero.mp hy with h | h\n · exact absurd h (ne_of_gt hnpos)\n · exact h\n linarith [hfy]\n show f a = f b\n rw [hconst a, hconst b]", "time_seconds": 122, "topic": "Markov chains (finite & countable)" }, { "question_id": 127, "problem_name": "stochastic_processes_scaled_branching_convergence", "informal_statement": "Consider a branching process (X_n) with i.i.d. offspring counts of mean Ȳ, and let Z_n = X_n/Ȳ^n be the scaled process.\n\nShow that Z_n is a nonnegative martingale and therefore converges with probability 1 to a finite limiting random variable.", "lean_statement": "theorem q127_scaled_branching_convergence (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hadap : Adapted ℱ X) (hint : ∀ n, Integrable (X n) μ)\n (hnn : ∀ n ω, 0 ≤ X n ω)\n (Ybar : ℝ) (hY : 0 < Ybar)\n (hbranch : ∀ n, μ[X (n + 1) | ℱ n] =ᵐ[μ] fun ω => Ybar * X n ω) :\n Martingale (fun n ω => X n ω / Ybar ^ n) ℱ μ ∧\n ∃ Zinf : Ω → ℝ, ∀ᵐ ω ∂μ, Tendsto (fun n => X n ω / Ybar ^ n) atTop (𝓝 (Zinf ω)) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q127_scaled_branching_convergence (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hadap : Adapted ℱ X) (hint : ∀ n, Integrable (X n) μ)\n (hnn : ∀ n ω, 0 ≤ X n ω)\n (Ybar : ℝ) (hY : 0 < Ybar)\n (hbranch : ∀ n, μ[X (n + 1) | ℱ n] =ᵐ[μ] fun ω => Ybar * X n ω) :\n Martingale (fun n ω => X n ω / Ybar ^ n) ℱ μ ∧\n ∃ Zinf : Ω → ℝ, ∀ᵐ ω ∂μ, Tendsto (fun n => X n ω / Ybar ^ n) atTop (𝓝 (Zinf ω)) := by\n set W : ℕ → Ω → ℝ := fun n ω => X n ω / Ybar ^ n with hW_def\n have hYne : Ybar ≠ 0 := ne_of_gt hY\n have hWadap : StronglyAdapted ℱ W := by\n intro i\n apply Measurable.stronglyMeasurable\n simp only [hW_def]\n exact (hadap i).div_const _\n have hWint : ∀ i, Integrable (W i) μ := by\n intro i\n simp only [hW_def]\n exact (hint i).div_const _\n have hWnn : ∀ n ω, 0 ≤ W n ω := by\n intro n ω\n simp only [hW_def]\n exact div_nonneg (hnn n ω) (pow_nonneg (le_of_lt hY) n)\n have hmart : Martingale W ℱ μ := by\n refine martingale_nat hWadap hWint ?_\n intro n\n set f : Ω → ℝ := fun _ => (Ybar ^ (n + 1))⁻¹ with hf_def\n set g : Ω → ℝ := fun ω => X (n + 1) ω with hg_def\n have hf_meas : StronglyMeasurable[ℱ n] f := stronglyMeasurable_const\n have hg_int : Integrable g μ := hint (n + 1)\n have hWeq : W (n + 1) = f * g := by\n funext ω\n simp only [hW_def, hf_def, hg_def, Pi.mul_apply]\n rw [div_eq_inv_mul]\n have hfg_int : Integrable (f * g) μ := by rw [← hWeq]; exact hWint (n + 1)\n have step1 : μ[W (n + 1) | ℱ n] =ᵐ[μ] f * μ[g | ℱ n] := by\n rw [hWeq]; exact condExp_mul_of_stronglyMeasurable_left hf_meas hfg_int hg_int\n have hgce : μ[g | ℱ n] =ᵐ[μ] fun ω => Ybar * X n ω := hbranch n\n have step2 : f * μ[g | ℱ n] =ᵐ[μ] W n := by\n filter_upwards [hgce] with ω hω\n simp only [Pi.mul_apply, hf_def, hW_def]\n rw [hω, pow_succ]\n field_simp\n exact (step1.trans step2).symm\n refine ⟨hmart, ℱ.limitProcess W μ, ?_⟩\n have hExp : ∀ n, ∫ ω, W n ω ∂μ = ∫ ω, W 0 ω ∂μ := by\n intro n\n have hce : μ[W n | ℱ 0] =ᵐ[μ] W 0 := hmart.2 0 n (Nat.zero_le n)\n calc ∫ ω, W n ω ∂μ = ∫ ω, (μ[W n | ℱ 0]) ω ∂μ := (integral_condExp (ℱ.le 0)).symm\n _ = ∫ ω, W 0 ω ∂μ := integral_congr_ae hce\n have hbound : ∀ n, eLpNorm (W n) 1 μ ≤ ((∫ ω, W 0 ω ∂μ).toNNReal : ℝ≥0∞) := by\n intro n\n rw [eLpNorm_one_eq_lintegral_enorm,\n ← ofReal_integral_norm_eq_lintegral_enorm (hWint n)]\n have hnorm : (∫ x, ‖W n x‖ ∂μ) = ∫ x, W n x ∂μ := by\n apply integral_congr_ae\n filter_upwards with ω\n rw [Real.norm_eq_abs, abs_of_nonneg (hWnn n ω)]\n rw [hnorm, hExp n]\n exact le_of_eq rfl\n exact hmart.submartingale.ae_tendsto_limitProcess hbound", "time_seconds": 85, "topic": "Martingales & stopping" }, { "question_id": 128, "problem_name": "stochastic_processes_drifted_brownian_martingale", "informal_statement": "Let B_μ(t) = μt + σB(t) be a Brownian motion with drift.\n\nShow that B_μ(t) − μt is a martingale, and that (B_μ(t) − μt)^2 − σ^2 t is a martingale.", "lean_statement": "theorem q128_drifted_brownian_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℝ m0) (B : ℝ → Ω → ℝ) (c σ : ℝ)\n (hadap : Adapted ℱ B)\n (hint : ∀ t, Integrable (B t) μ)\n (hint2 : ∀ t, Integrable (fun ω => (B t ω) ^ 2) μ)\n (hB0 : ∀ᵐ ω ∂μ, B 0 ω = 0)\n (hincr : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n HasLaw (fun ω => B t ω - B s ω) (gaussianReal 0 (t - s).toNNReal) μ)\n (hpast : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n Indep (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (ℱ s) μ) :\n (∀ s t : ℝ, 0 ≤ s → s ≤ t →\n μ[(fun ω => (c * t + σ * B t ω) - c * t) | (ℱ s)]\n =ᵐ[μ] (fun ω => (c * s + σ * B s ω) - c * s)) ∧\n (∀ s t : ℝ, 0 ≤ s → s ≤ t →\n μ[(fun ω => ((c * t + σ * B t ω) - c * t) ^ 2 - σ ^ 2 * t) | (ℱ s)]\n =ᵐ[μ] (fun ω => ((c * s + σ * B s ω) - c * s) ^ 2 - σ ^ 2 * s)) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q128_drifted_brownian_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℝ m0) (B : ℝ → Ω → ℝ) (c σ : ℝ)\n (hadap : Adapted ℱ B)\n (hint : ∀ t, Integrable (B t) μ)\n (hint2 : ∀ t, Integrable (fun ω => (B t ω) ^ 2) μ)\n (hB0 : ∀ᵐ ω ∂μ, B 0 ω = 0)\n (hincr : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n HasLaw (fun ω => B t ω - B s ω) (gaussianReal 0 (t - s).toNNReal) μ)\n (hpast : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n Indep (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (ℱ s) μ) :\n (∀ s t : ℝ, 0 ≤ s → s ≤ t →\n μ[(fun ω => (c * t + σ * B t ω) - c * t) | (ℱ s)]\n =ᵐ[μ] (fun ω => (c * s + σ * B s ω) - c * s)) ∧\n (∀ s t : ℝ, 0 ≤ s → s ≤ t →\n μ[(fun ω => ((c * t + σ * B t ω) - c * t) ^ 2 - σ ^ 2 * t) | (ℱ s)]\n =ᵐ[μ] (fun ω => ((c * s + σ * B s ω) - c * s) ^ 2 - σ ^ 2 * s)) := by\n have hsq_int : ∀ v : NNReal, ∫ x, x ^ 2 ∂(gaussianReal (0:ℝ) v) = (v : ℝ) := by\n intro v\n have hv := variance_of_integral_eq_zero (μ := gaussianReal (0:ℝ) v) (X := fun x => x)\n measurable_id.aemeasurable (by simpa using integral_id_gaussianReal (μ := (0:ℝ)) (v := v))\n rw [variance_fun_id_gaussianReal] at hv\n simpa using hv.symm\n have main : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n (μ[B t | ℱ s] =ᵐ[μ] B s) ∧\n (μ[fun ω => (B t ω) ^ 2 | ℱ s] =ᵐ[μ] fun ω => (B s ω) ^ 2 + (t - s)) := by\n intro s t hs hst\n have hst0 : (0:ℝ) ≤ t - s := by linarith\n have hBt : Measurable (B t) := (hadap t).mono (ℱ.le t) le_rfl\n have hBs : Measurable (B s) := (hadap s).mono (ℱ.le s) le_rfl\n have hg_meas : Measurable (fun ω => B t ω - B s ω) := hBt.sub hBs\n have hg_int : Integrable (fun ω => B t ω - B s ω) μ := (hint t).sub (hint s)\n have hindep := hpast s t hs hst\n have hle1 : MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance ≤ m0 :=\n hg_meas.comap_le\n have hle2 : ℱ s ≤ m0 := ℱ.le s\n have hsm : StronglyMeasurable[MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance]\n (fun ω => B t ω - B s ω) := (measurable_iff_comap_le.2 le_rfl).stronglyMeasurable\n have hEg : μ[fun ω => B t ω - B s ω] = 0 := by\n rw [(hincr s t hs hst).integral_eq, integral_id_gaussianReal]\n have hcz : μ[fun ω => B t ω - B s ω | ℱ s] =ᵐ[μ] fun _ => (0 : ℝ) := by\n have h := condExp_indep_eq hle1 hle2 hsm hindep\n rw [hEg] at h; exact h\n have hmart : μ[B t | ℱ s] =ᵐ[μ] B s := by\n have hcond_Bs : μ[B s | ℱ s] = B s :=\n condExp_of_stronglyMeasurable hle2 (hadap s).stronglyMeasurable (hint s)\n have key : (B t) = (B s) + (fun ω => B t ω - B s ω) := by\n funext ω; show B t ω = B s ω + (B t ω - B s ω); ring\n have hadd : μ[B t | ℱ s] =ᵐ[μ]\n (μ[B s | ℱ s] + μ[fun ω => B t ω - B s ω | ℱ s]) := by\n calc μ[B t | ℱ s] = μ[(B s) + (fun ω => B t ω - B s ω) | ℱ s] :=\n congrArg (fun h => μ[h | ℱ s]) key\n _ =ᵐ[μ] _ := condExp_add (hint s) hg_int _\n filter_upwards [hadd, hcz] with ω h1 h3\n rw [h1]; simp [Pi.add_apply, hcond_Bs, h3]\n refine ⟨hmart, ?_⟩\n have hmemBt : MemLp (B t) 2 μ :=\n (memLp_two_iff_integrable_sq hBt.aestronglyMeasurable).2 (hint2 t)\n have hmemBs : MemLp (B s) 2 μ :=\n (memLp_two_iff_integrable_sq hBs.aestronglyMeasurable).2 (hint2 s)\n have hmemInc : MemLp (fun ω => B t ω - B s ω) 2 μ := hmemBt.sub hmemBs\n have hd_int : Integrable (fun ω => (B t ω - B s ω) ^ 2) μ := hmemInc.integrable_sq\n have hmem2Bs : MemLp (fun ω => 2 * B s ω) 2 μ := hmemBs.const_mul 2\n have hb_int : Integrable ((fun ω => 2 * B s ω) * (fun ω => B t ω - B s ω)) μ :=\n hmem2Bs.integrable_mul hmemInc\n have ha_int : Integrable (fun ω => (B s ω) ^ 2) μ := hint2 s\n have ea : μ[fun ω => (B s ω) ^ 2 | ℱ s] = fun ω => (B s ω) ^ 2 :=\n condExp_of_stronglyMeasurable hle2 ((hadap s).pow_const 2).stronglyMeasurable ha_int\n have eb : μ[(fun ω => 2 * B s ω) * (fun ω => B t ω - B s ω) | ℱ s] =ᵐ[μ] fun _ => (0:ℝ) := by\n have hpull := condExp_mul_of_stronglyMeasurable_left\n (((hadap s).const_mul 2).stronglyMeasurable) hb_int hg_int\n filter_upwards [hpull, hcz] with ω h1 h3\n rw [h1]; simp [Pi.mul_apply, h3]\n have hsm_d : StronglyMeasurable[MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance]\n (fun ω => (B t ω - B s ω) ^ 2) :=\n ((measurable_iff_comap_le.2 le_rfl).pow_const 2).stronglyMeasurable\n have hEd : μ[fun ω => (B t ω - B s ω) ^ 2] = t - s := by\n have hc := (hincr s t hs hst).integral_comp (f := fun x : ℝ => x ^ 2) (by fun_prop)\n simp only [Function.comp_def] at hc\n rw [hc, hsq_int, Real.coe_toNNReal _ hst0]\n have ed : μ[fun ω => (B t ω - B s ω) ^ 2 | ℱ s] =ᵐ[μ] fun _ => (t - s) := by\n have h := condExp_indep_eq hle1 hle2 hsm_d hindep\n rw [hEd] at h; exact h\n have key2 : (fun ω => (B t ω) ^ 2) =\n (fun ω => (B s ω) ^ 2) + (fun ω => 2 * B s ω) * (fun ω => B t ω - B s ω)\n + (fun ω => (B t ω - B s ω) ^ 2) := by\n funext ω; simp only [Pi.add_apply, Pi.mul_apply]; ring\n have hcongr : μ[fun ω => (B t ω) ^ 2 | ℱ s] =\n μ[(fun ω => (B s ω) ^ 2) + (fun ω => 2 * B s ω) * (fun ω => B t ω - B s ω)\n + (fun ω => (B t ω - B s ω) ^ 2) | ℱ s] :=\n congrArg (fun h => μ[h | ℱ s]) key2\n have e2 : μ[(fun ω => (B s ω) ^ 2) + (fun ω => 2 * B s ω) * (fun ω => B t ω - B s ω) | ℱ s]\n =ᵐ[μ] (μ[fun ω => (B s ω) ^ 2 | ℱ s]\n + μ[(fun ω => 2 * B s ω) * (fun ω => B t ω - B s ω) | ℱ s]) :=\n condExp_add ha_int hb_int _\n have e1 : μ[(fun ω => (B s ω) ^ 2) + (fun ω => 2 * B s ω) * (fun ω => B t ω - B s ω)\n + (fun ω => (B t ω - B s ω) ^ 2) | ℱ s]\n =ᵐ[μ] (μ[(fun ω => (B s ω) ^ 2) + (fun ω => 2 * B s ω) * (fun ω => B t ω - B s ω) | ℱ s]\n + μ[fun ω => (B t ω - B s ω) ^ 2 | ℱ s]) :=\n condExp_add (ha_int.add hb_int) hd_int _\n rw [hcongr]\n filter_upwards [e1, e2, eb, ed] with ω h1 h2 hb hd\n rw [h1]\n simp only [Pi.add_apply] at h2 ⊢\n rw [h2, ea]\n simp only [Pi.add_apply] at hb hd ⊢\n rw [hb, hd]\n ring\n constructor\n · intro s t hs hst\n have h368 := (main s t hs hst).1\n have harg : (fun ω => (c * t + σ * B t ω) - c * t) = σ • B t := by\n funext ω; simp only [Pi.smul_apply, smul_eq_mul]; ring\n rw [harg]\n have hsmul := condExp_smul (μ := μ) σ (B t) (ℱ s : MeasurableSpace Ω)\n filter_upwards [hsmul, h368] with ω h1 h2\n simp only [Pi.smul_apply, smul_eq_mul] at h1\n rw [h1, h2]; ring\n · intro s t hs hst\n have hquad := (main s t hs hst).2\n have harg : (fun ω => ((c * t + σ * B t ω) - c * t) ^ 2 - σ ^ 2 * t) =\n (σ ^ 2 • (fun ω => (B t ω) ^ 2)) - (fun _ => σ ^ 2 * t) := by\n funext ω; simp only [Pi.sub_apply, Pi.smul_apply, smul_eq_mul]; ring\n rw [harg]\n have hconst : μ[(fun _ => σ ^ 2 * t) | ℱ s] = fun _ => σ ^ 2 * t :=\n condExp_const (ℱ.le s) _\n have hsmul := condExp_smul (μ := μ) (σ ^ 2) (fun ω => (B t ω) ^ 2) (ℱ s : MeasurableSpace Ω)\n have hBt2 : MemLp (B t) 2 μ :=\n (memLp_two_iff_integrable_sq\n (((hadap t).mono (ℱ.le t) le_rfl).aestronglyMeasurable)).2 (hint2 t)\n have hsub : μ[(σ ^ 2 • (fun ω => (B t ω) ^ 2)) - (fun _ => σ ^ 2 * t) | ℱ s]\n =ᵐ[μ] (μ[σ ^ 2 • (fun ω => (B t ω) ^ 2) | ℱ s] - μ[(fun _ => σ ^ 2 * t) | ℱ s]) :=\n condExp_sub (Integrable.smul (σ ^ 2) hBt2.integrable_sq) (integrable_const _) _\n filter_upwards [hsub, hsmul, hquad] with ω hs1 hs2 hq\n rw [hs1]\n simp only [Pi.sub_apply, hconst]\n rw [hs2]\n simp only [Pi.smul_apply, smul_eq_mul]\n rw [hq]\n ring", "time_seconds": 846, "topic": "Brownian motion & stochastic calculus" }, { "question_id": 129, "problem_name": "stochastic_processes_random_walk_recurrent_1", "informal_statement": "Show that simple random walk on ℤ^2 is recurrent.", "lean_statement": "theorem q129_z2_recurrent :\n ¬ Summable (fun n : ℕ => ((Nat.choose (2 * n) n : ℝ) / (4 : ℝ) ^ n) ^ 2) := by sorry", "true_to_source": "abstract", "proof": "theorem q129_z2_recurrent :\n ¬ Summable (fun n : ℕ => ((Nat.choose (2 * n) n : ℝ) / (4 : ℝ) ^ n) ^ 2) := by\n intro hsum\n set a : ℕ → ℝ := fun n => (Nat.choose (2 * n) n : ℝ) / (4 : ℝ) ^ n with ha\n -- a n ≥ 0\n have hapos : ∀ n, 0 ≤ a n := by\n intro n; rw [ha]; positivity\n -- recurrence: (n+1) * a (n+1) = (2n+1)/2 * a n (in the form 2(n+1) a(n+1) = (2n+1) a n)\n have hrec : ∀ n : ℕ, (2 * ((n : ℝ) + 1)) * a (n + 1) = (2 * (n : ℝ) + 1) * a n := by\n intro n\n have hk := Nat.succ_mul_centralBinom_succ n\n rw [Nat.centralBinom_eq_two_mul_choose, Nat.centralBinom_eq_two_mul_choose] at hk\n have hkR : ((n : ℝ) + 1) * (Nat.choose (2 * (n + 1)) (n + 1) : ℝ)\n = 2 * (2 * (n : ℝ) + 1) * (Nat.choose (2 * n) n : ℝ) := by\n have : ((((n + 1) * (2 * (n + 1)).choose (n + 1) : ℕ)) : ℝ)\n = (((2 * (2 * n + 1) * (2 * n).choose n : ℕ)) : ℝ) := by exact_mod_cast hk\n push_cast at this ⊢\n linarith [this]\n rw [ha]\n simp only\n have h4 : (4 : ℝ) ^ (n + 1) = 4 * 4 ^ n := by rw [pow_succ]; ring\n rw [h4]\n have h4pos : (0 : ℝ) < (4 : ℝ) ^ n := by positivity\n field_simp\n ring_nf\n ring_nf at hkR\n linarith [hkR]\n -- square bound: 1/(4n) ≤ (a n)^2 for n ≥ 1\n have hsq : ∀ n : ℕ, 1 ≤ n → 1 / (4 * (n : ℝ)) ≤ (a n) ^ 2 := by\n intro n hn\n induction n with\n | zero => omega\n | succ k ih =>\n rcases Nat.eq_zero_or_pos k with hk0 | hk0\n · subst hk0\n -- a 1 = 2/4 = 1/2, (a 1)^2 = 1/4 = 1/(4*1)\n rw [ha]\n norm_num [Nat.choose]\n · have ihk := ih hk0\n -- a (k+1) = (2k+1)/(2(k+1)) * a k\n have hkR1 : (0 : ℝ) < 2 * ((k : ℝ) + 1) := by positivity\n have hreck := hrec k\n have haeq : a (k + 1) = (2 * (k : ℝ) + 1) / (2 * ((k : ℝ) + 1)) * a k := by\n rw [div_mul_eq_mul_div, eq_div_iff (ne_of_gt hkR1)]\n linarith [hreck]\n rw [haeq]\n rw [mul_pow]\n have hkpos : (0 : ℝ) < (k : ℝ) := by exact_mod_cast hk0\n -- ((2k+1)/(2(k+1)))^2 * (a k)^2 ≥ ((2k+1)/(2(k+1)))^2 * (1/(4k))\n have hfac : (0 : ℝ) ≤ ((2 * (k : ℝ) + 1) / (2 * ((k : ℝ) + 1))) ^ 2 := by positivity\n have step1 : ((2 * (k : ℝ) + 1) / (2 * ((k : ℝ) + 1))) ^ 2 * (1 / (4 * (k : ℝ)))\n ≤ ((2 * (k : ℝ) + 1) / (2 * ((k : ℝ) + 1))) ^ 2 * (a k) ^ 2 :=\n mul_le_mul_of_nonneg_left ihk hfac\n refine le_trans ?_ step1\n -- need: 1/(4(k+1)) ≤ ((2k+1)/(2(k+1)))^2 * (1/(4k))\n have hlhs : (0 : ℝ) < 4 * ((k : ℝ) + 1) := by positivity\n have hrhs : (0 : ℝ) < 4 * (k : ℝ) * (2 * ((k : ℝ) + 1)) ^ 2 := by positivity\n have hcast : (4 : ℝ) * ((k : ℕ) + 1 : ℕ) = 4 * ((k : ℝ) + 1) := by push_cast; ring\n rw [div_pow, div_mul_eq_mul_div, mul_one_div, div_div, hcast]\n rw [div_le_div_iff₀ hlhs hrhs]\n nlinarith [sq_nonneg ((k : ℝ)), hkpos]\n -- comparison: ∑ 1/(4n) ≤ ∑ (a n)^2 ⟹ 1/n summable, contradiction\n have hcomp : Summable (fun n : ℕ => 1 / (4 * (n : ℝ))) := by\n apply Summable.of_nonneg_of_le (g := fun n : ℕ => 1 / (4 * (n : ℝ)))\n (f := fun n => (a n) ^ 2)\n · intro n; positivity\n · intro n\n rcases Nat.eq_zero_or_pos n with hn | hn\n · subst hn; simp; positivity\n · exact hsq n hn\n · exact hsum\n have : Summable (fun n : ℕ => 1 / (n : ℝ)) := by\n have h4 : Summable (fun n : ℕ => 4 * (1 / (4 * (n : ℝ)))) := hcomp.mul_left 4\n apply h4.congr\n intro n\n rcases Nat.eq_zero_or_pos n with hn | hn\n · subst hn; simp\n · field_simp\n exact Real.not_summable_one_div_natCast this", "time_seconds": 83, "topic": "Random walks & large deviations" }, { "question_id": 131, "problem_name": "stochastic_processes_markov_modulated_threshold_bound", "informal_statement": "Let a Markov-modulated random walk have negative drift and let r* > 0 satisfy ρ(r*) = 1, where ρ(r) is the largest eigenvalue of [A(r)], with positive right eigenvector ν(r*).\n\nShow that the probability the walk ever crosses a threshold α > 0 is bounded by a constant times exp(−r* α).", "lean_statement": "theorem q131_markov_modulated_threshold_bound {σ : Type*} [Fintype σ]\n (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (Y : ℕ → Ω → σ) (S : ℕ → Ω → ℝ) (rstar : ℝ) (hr : 0 < rstar)\n (ν : σ → ℝ) (νmin : ℝ) (hνmin : 0 < νmin) (hν : ∀ s, νmin ≤ ν s)\n (Z : ℕ → Ω → ℝ) (hZ : ∀ n ω, Z n ω = Real.exp (rstar * S n ω) * ν (Y n ω))\n (hZmart : Martingale Z ℱ μ) (hZnn : ∀ n ω, 0 ≤ Z n ω) :\n ∃ C : ℝ, 0 < C ∧\n ∀ α : ℝ, 0 < α → (μ {ω | ∃ n, α ≤ S n ω}).toReal ≤ C * Real.exp (-rstar * α) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q131_markov_modulated_threshold_bound {σ : Type*} [Fintype σ]\n (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (Y : ℕ → Ω → σ) (S : ℕ → Ω → ℝ) (rstar : ℝ) (hr : 0 < rstar)\n (ν : σ → ℝ) (νmin : ℝ) (hνmin : 0 < νmin) (hν : ∀ s, νmin ≤ ν s)\n (Z : ℕ → Ω → ℝ) (hZ : ∀ n ω, Z n ω = Real.exp (rstar * S n ω) * ν (Y n ω))\n (hZmart : Martingale Z ℱ μ) (hZnn : ∀ n ω, 0 ≤ Z n ω) :\n ∃ C : ℝ, 0 < C ∧\n ∀ α : ℝ, 0 < α → (μ {ω | ∃ n, α ≤ S n ω}).toReal ≤ C * Real.exp (-rstar * α) := by\n -- 183-style nonnegative martingale maximal inequality for Z, for every threshold a > 0\n have h183 : ∀ a : ℝ, 0 < a →\n (μ {ω | ∃ n, a ≤ Z n ω}).toReal ≤ (∫ ω, Z 0 ω ∂μ) / a := by\n intro a ha\n have hrm : ∀ n ω, runningMax Z n ω\n = (Finset.range (n+1)).sup' (by simp) (fun k => Z k ω) := by\n intro n ω\n unfold runningMax\n rw [← Finset.sup'_univ_eq_ciSup]\n apply le_antisymm\n · apply Finset.sup'_le; intro k _\n apply Finset.le_sup' (fun k => Z k ω); simp [Finset.mem_range, k.isLt]\n · apply Finset.sup'_le; intro j hj\n rw [Finset.mem_range] at hj\n exact Finset.le_sup' (fun k : Fin (n+1) => Z (↑k) ω) (Finset.mem_univ ⟨j, hj⟩)\n have hcoe : ((a.toNNReal : ℝ≥0) : ℝ) = a := Real.coe_toNNReal a ha.le\n have hsub : Submartingale Z ℱ μ := hZmart.submartingale\n have hnn' : (0 : ℕ → Ω → ℝ) ≤ Z := fun n ω => hZnn n ω\n have hstep : ∀ n, (μ {ω | a ≤ runningMax Z n ω}).toReal ≤ (∫ ω, Z 0 ω ∂μ) / a := by\n intro n\n have key := MeasureTheory.maximal_ineq hsub hnn' (ε := a.toNNReal) n\n rw [hcoe] at key\n simp only [← hrm] at key\n have hI : 0 ≤ ∫ ω in {ω | a ≤ runningMax Z n ω}, Z n ω ∂μ :=\n integral_nonneg (fun ω => hZnn n ω)\n have key2 := ENNReal.toReal_mono ENNReal.ofReal_ne_top key\n rw [ENNReal.toReal_ofReal hI, ENNReal.toReal_mul, ENNReal.coe_toReal, hcoe] at key2\n have hsubint : ∫ ω in {ω | a ≤ runningMax Z n ω}, Z n ω ∂μ ≤ ∫ ω, Z n ω ∂μ :=\n setIntegral_le_integral (hsub.integrable n) (ae_of_all _ (fun ω => hZnn n ω))\n have hEn : ∫ ω, Z n ω ∂μ = ∫ ω, Z 0 ω ∂μ := by\n have := hZmart.setIntegral_eq (i := 0) (j := n) (Nat.zero_le n) (s := Set.univ)\n MeasurableSet.univ\n simpa using this.symm\n rw [hEn] at hsubint\n have hfin : a * (μ {ω | a ≤ runningMax Z n ω}).toReal ≤ ∫ ω, Z 0 ω ∂μ :=\n le_trans key2 hsubint\n rw [le_div_iff₀ ha, mul_comm]\n exact hfin\n have hUnion : {ω | ∃ n, a ≤ Z n ω} = ⋃ n, {ω | a ≤ runningMax Z n ω} := by\n ext ω\n simp only [Set.mem_setOf_eq, Set.mem_iUnion]\n constructor\n · rintro ⟨n, hn⟩\n refine ⟨n, ?_⟩\n rw [hrm n ω]\n exact le_trans hn\n (Finset.le_sup' (fun k => Z k ω) (Finset.mem_range.mpr (Nat.lt_succ_self n)))\n · rintro ⟨n, hn⟩\n rw [hrm n ω, Finset.le_sup'_iff] at hn\n obtain ⟨k, _, hk⟩ := hn\n exact ⟨k, hk⟩\n have hmono : Monotone (fun n => {ω | a ≤ runningMax Z n ω}) := by\n intro i j hij ω hω\n simp only [Set.mem_setOf_eq] at hω ⊢\n rw [hrm i ω, Finset.le_sup'_iff] at hω\n rw [hrm j ω, Finset.le_sup'_iff]\n obtain ⟨k, hk, hk2⟩ := hω\n exact ⟨k, Finset.mem_range.mpr\n (lt_of_lt_of_le (Finset.mem_range.mp hk) (Nat.succ_le_succ hij)), hk2⟩\n have htend := tendsto_measure_iUnion_atTop (μ := μ) hmono\n have hc0 : 0 ≤ (∫ ω, Z 0 ω ∂μ) / a :=\n div_nonneg (integral_nonneg (fun ω => hZnn 0 ω)) ha.le\n have hbound : ∀ n, μ {ω | a ≤ runningMax Z n ω} ≤ ENNReal.ofReal ((∫ ω, Z 0 ω ∂μ)/a) := by\n intro n\n rw [← ENNReal.ofReal_toReal (measure_ne_top μ _)]\n exact ENNReal.ofReal_le_ofReal (hstep n)\n have hlim : μ (⋃ n, {ω | a ≤ runningMax Z n ω}) ≤ ENNReal.ofReal ((∫ ω, Z 0 ω ∂μ)/a) :=\n le_of_tendsto htend (Filter.Eventually.of_forall hbound)\n rw [hUnion]\n calc (μ (⋃ n, {ω | a ≤ runningMax Z n ω})).toReal\n ≤ (ENNReal.ofReal ((∫ ω, Z 0 ω ∂μ)/a)).toReal :=\n ENNReal.toReal_mono ENNReal.ofReal_ne_top hlim\n _ = (∫ ω, Z 0 ω ∂μ)/a := ENNReal.toReal_ofReal hc0\n -- Now the Chernoff-type bound\n have hint0 : 0 ≤ ∫ ω, Z 0 ω ∂μ := integral_nonneg (fun ω => hZnn 0 ω)\n refine ⟨(∫ ω, Z 0 ω ∂μ) / νmin + 1, ?_, ?_⟩\n · have : 0 ≤ (∫ ω, Z 0 ω ∂μ) / νmin := div_nonneg hint0 hνmin.le\n linarith\n · intro α hα\n set a : ℝ := Real.exp (rstar * α) * νmin with ha_def\n have ha : 0 < a := mul_pos (Real.exp_pos _) hνmin\n -- set inclusion\n have hincl : {ω | ∃ n, α ≤ S n ω} ⊆ {ω | ∃ n, a ≤ Z n ω} := by\n rintro ω ⟨n, hn⟩\n refine ⟨n, ?_⟩\n rw [hZ n ω, ha_def]\n have h1 : Real.exp (rstar * α) ≤ Real.exp (rstar * S n ω) :=\n Real.exp_le_exp.mpr (by nlinarith [hn, hr])\n have h2 : νmin ≤ ν (Y n ω) := hν _\n calc Real.exp (rstar * α) * νmin\n ≤ Real.exp (rstar * S n ω) * νmin :=\n mul_le_mul_of_nonneg_right h1 hνmin.le\n _ ≤ Real.exp (rstar * S n ω) * ν (Y n ω) :=\n mul_le_mul_of_nonneg_left h2 (Real.exp_pos _).le\n have hmono_meas : (μ {ω | ∃ n, α ≤ S n ω}).toReal ≤ (μ {ω | ∃ n, a ≤ Z n ω}).toReal :=\n ENNReal.toReal_mono (measure_ne_top μ _) (measure_mono hincl)\n have hkey : (μ {ω | ∃ n, a ≤ Z n ω}).toReal ≤ (∫ ω, Z 0 ω ∂μ) / a := h183 a ha\n -- (∫ Z0)/a = ((∫ Z0)/νmin) * exp(-rstar*α)\n have hrw : (∫ ω, Z 0 ω ∂μ) / a = ((∫ ω, Z 0 ω ∂μ) / νmin) * Real.exp (-rstar * α) := by\n rw [ha_def]\n rw [show -rstar * α = -(rstar * α) by ring, Real.exp_neg]\n field_simp\n have hCbound : ((∫ ω, Z 0 ω ∂μ) / νmin) * Real.exp (-rstar * α)\n ≤ ((∫ ω, Z 0 ω ∂μ) / νmin + 1) * Real.exp (-rstar * α) :=\n mul_le_mul_of_nonneg_right (by linarith) (Real.exp_pos _).le\n calc (μ {ω | ∃ n, α ≤ S n ω}).toReal\n ≤ (μ {ω | ∃ n, a ≤ Z n ω}).toReal := hmono_meas\n _ ≤ (∫ ω, Z 0 ω ∂μ) / a := hkey\n _ = ((∫ ω, Z 0 ω ∂μ) / νmin) * Real.exp (-rstar * α) := hrw\n _ ≤ ((∫ ω, Z 0 ω ∂μ) / νmin + 1) * Real.exp (-rstar * α) := hCbound", "time_seconds": 207, "topic": "Random walks & large deviations" }, { "question_id": 132, "problem_name": "stochastic_processes_mm1_expected_queue_delay", "informal_statement": "Consider an M/M/1 queue in steady state with arrival rate λ, service rate μ, and ρ = λ/μ < 1.\n\nShow that the expected waiting time in queue (excluding service) is ρ/(μ − λ).", "lean_statement": "theorem q132_mm1_expected_queue_delay\n (lam mu : ℝ) (hlam : 0 < lam) (hmu : 0 < mu) (hstab : lam < mu)\n (ρ : ℝ) (hρ : ρ = lam / mu) (hρ0 : 0 < ρ) (hρ1 : ρ < 1)\n (p : ℕ → ℝ) (hp : ∀ i, p i = (1 - ρ) * ρ ^ i)\n (Wq Lq : ℝ) (hLq : Lq = ∑' k : ℕ, (k : ℝ) * p (k + 1))\n (hLittle : Lq = lam * Wq) :\n Wq = ρ / (mu - lam) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q132_mm1_expected_queue_delay\n (lam mu : ℝ) (hlam : 0 < lam) (hmu : 0 < mu) (hstab : lam < mu)\n (ρ : ℝ) (hρ : ρ = lam / mu) (hρ0 : 0 < ρ) (hρ1 : ρ < 1)\n (p : ℕ → ℝ) (hp : ∀ i, p i = (1 - ρ) * ρ ^ i)\n (Wq Lq : ℝ) (hLq : Lq = ∑' k : ℕ, (k : ℝ) * p (k + 1))\n (hLittle : Lq = lam * Wq) :\n Wq = ρ / (mu - lam) := by\n have hlamne : lam ≠ 0 := hlam.ne'\n have h1ρpos : (0:ℝ) < 1 - ρ := by linarith\n have h1ρ : (1:ℝ) - ρ ≠ 0 := h1ρpos.ne'\n have hmlpos : (0:ℝ) < mu - lam := by linarith\n have hgeo : ∑' k : ℕ, (k : ℝ) * ρ ^ k = ρ / (1 - ρ) ^ 2 := by\n have hnorm : ‖ρ‖ < 1 := by rw [Real.norm_eq_abs, abs_of_pos hρ0]; exact hρ1\n exact tsum_coe_mul_geometric_of_norm_lt_one hnorm\n have hLqval : Lq = ρ ^ 2 / (1 - ρ) := by\n rw [hLq]\n have hterm : ∀ n : ℕ, (n : ℝ) * p (n + 1) = ((1 - ρ) * ρ) * ((n : ℝ) * ρ ^ n) := by\n intro n; rw [hp, pow_succ]; ring\n rw [tsum_congr hterm, tsum_mul_left, hgeo]\n field_simp\n have hWeq : lam * Wq = ρ ^ 2 / (1 - ρ) := hLittle ▸ hLqval\n have hWeq' : lam * Wq * (1 - ρ) = ρ ^ 2 := by\n rw [hWeq]; field_simp\n have hcore : ρ * (mu - lam) = (1 - ρ) * lam := by\n rw [hρ]; field_simp\n rw [eq_div_iff hmlpos.ne']\n have hne : lam * (1 - ρ) ≠ 0 := mul_ne_zero hlamne h1ρ\n apply mul_left_cancel₀ hne\n linear_combination (mu - lam) * hWeq' + ρ * hcore", "time_seconds": 214, "topic": "Continuous-time Markov & queues" }, { "question_id": 136, "problem_name": "stochastic_processes_three_state_cycle_nonreversible", "informal_statement": "Let (X_n) be an irreducible positive-recurrent Markov chain with stationary distribution π containing three states i, j, k.\n\nShow that the chain is reversible only if P_{ij} P_{jk} P_{ki} = P_{ik} P_{kj} P_{ji}; if this product condition fails on any cycle, the chain is not reversible.", "lean_statement": "theorem q136_three_state_cycle\n {S : Type*}\n (P : S → S → ℝ) (π : S → ℝ) (hπpos : ∀ s, 0 < π s)\n (hDB : ∀ a b, π a * P a b = π b * P b a)\n (i j k : S) :\n P i j * P j k * P k i = P i k * P k j * P j i := by\n sorry", "true_to_source": "abstract", "proof": "theorem q136_three_state_cycle\n {S : Type*}\n (P : S → S → ℝ) (π : S → ℝ) (hπpos : ∀ s, 0 < π s)\n (hDB : ∀ a b, π a * P a b = π b * P b a)\n (i j k : S) :\n P i j * P j k * P k i = P i k * P k j * P j i := by\n have h1 := hDB i j\n have h2 := hDB j k\n have h3 := hDB k i\n have hpos : π i * π j * π k ≠ 0 :=\n ne_of_gt (mul_pos (mul_pos (hπpos i) (hπpos j)) (hπpos k))\n have e : π i * π j * π k * (P i j * P j k * P k i)\n = π i * π j * π k * (P i k * P k j * P j i) := by\n calc π i * π j * π k * (P i j * P j k * P k i)\n = (π i * P i j) * (π j * P j k) * (π k * P k i) := by ring\n _ = (π j * P j i) * (π k * P k j) * (π i * P i k) := by rw [h1, h2, h3]\n _ = π i * π j * π k * (P i k * P k j * P j i) := by ring\n exact mul_left_cancel₀ hpos e", "time_seconds": 49, "topic": "Markov chains (finite & countable)" }, { "question_id": 140, "problem_name": "stochastic_processes_birth_death_null_recurrent_boundary", "informal_statement": "Consider an irreducible birth-death chain on the nonnegative integers with ρ_i = p_i/q_{i+1} and ∏_{j 1 / (p n * ∏ j ∈ Finset.range n, ρ j)))\n ∧ (¬ Summable (fun n : ℕ => ∏ j ∈ Finset.range n, ρ j)) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q140_birth_death_null_recurrent_boundary\n (p q : ℕ → ℝ)\n (hp : ∀ i, 0 < p i) (hq0 : q 0 = 0) (hqpos : ∀ i, 0 < q (i + 1))\n (hbound : ∀ i, p i + q i ≤ 1)\n (ρ : ℕ → ℝ) (hρ : ∀ i, ρ i = p i / q (i + 1))\n (hprod : ∀ i, ∏ j ∈ Finset.range i, ρ j = 1) :\n (¬ Summable (fun n : ℕ => 1 / (p n * ∏ j ∈ Finset.range n, ρ j)))\n ∧ (¬ Summable (fun n : ℕ => ∏ j ∈ Finset.range n, ρ j)) := by\n have key : ∀ n, ∏ j ∈ Finset.range n, ρ j = 1 := hprod\n have hple : ∀ n, p n ≤ 1 := by\n intro n\n have hq : 0 ≤ q n := by\n cases n with\n | zero => rw [hq0]\n | succ k => exact (hqpos k).le\n linarith [hbound n, hp n]\n constructor\n · intro hsum\n have hsum' : Summable (fun n : ℕ => 1 / p n) := by\n have he : (fun n : ℕ => 1 / (p n * ∏ j ∈ Finset.range n, ρ j)) = (fun n => 1 / p n) := by\n funext n; rw [key n, mul_one]\n rwa [he] at hsum\n have htend := hsum'.tendsto_atTop_zero\n have hge : ∀ n, (1:ℝ) ≤ 1 / p n := by\n intro n\n rw [le_div_iff₀ (hp n)]\n linarith [hple n]\n have : (1:ℝ) ≤ 0 := ge_of_tendsto' htend hge\n linarith\n · intro hsum\n have hsum' : Summable (fun _ : ℕ => (1:ℝ)) := by\n have he : (fun n : ℕ => ∏ j ∈ Finset.range n, ρ j) = (fun _ => (1:ℝ)) := by\n funext n; exact key n\n rwa [he] at hsum\n have htend := hsum'.tendsto_atTop_zero\n have : (1:ℝ) = 0 := tendsto_nhds_unique tendsto_const_nhds htend\n norm_num at this", "time_seconds": 131, "topic": "Markov chains (finite & countable)" }, { "question_id": 141, "problem_name": "stochastic_processes_submartingale_1", "informal_statement": "Let (X_n)_{n ≥ 0} be a martingale, and let ϕ be a convex function.\n\n Show that (ϕ(X_n))_{n ≥ 0} is a submartingale.", "lean_statement": "theorem q141_convex_submartingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hX : Martingale X ℱ μ) (φ : ℝ → ℝ) (hconv : ConvexOn ℝ Set.univ φ)\n (hint : ∀ n, Integrable (fun ω => φ (X n ω)) μ) :\n Submartingale (fun n ω => φ (X n ω)) ℱ μ := by sorry", "true_to_source": "literal", "proof": "theorem q141_convex_submartingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hX : Martingale X ℱ μ) (φ : ℝ → ℝ) (hconv : ConvexOn ℝ Set.univ φ)\n (hint : ∀ n, Integrable (fun ω => φ (X n ω)) μ) :\n Submartingale (fun n ω => φ (X n ω)) ℱ μ := by\n have hcont : Continuous φ := hconv.locallyLipschitz.continuous\n refine ⟨fun n => hcont.comp_stronglyMeasurable (hX.stronglyMeasurable n), ?_, hint⟩\n intro i j hij\n have hmart := hX.condExp_ae_eq hij\n have hjensen := hconv.map_condExp_le_univ (ℱ.le i) hcont.lowerSemicontinuous\n (hX.integrable j) (hint j)\n filter_upwards [hjensen, hmart] with ω hω hω2\n simp only [Function.comp_apply] at hω\n rw [hω2] at hω\n exact hω", "time_seconds": 304, "topic": "Martingales & stopping" }, { "question_id": 143, "problem_name": "stochastic_processes_convergence_theorem_2", "informal_statement": "Suppose that (X_n)_n is a Markov chain with transition matrix P.\n Assume that P is irreducible and aperiodic.\n\n Show that there exists r such that P^r (x, y) > 0 for all x,y ∈ Ω.", "lean_statement": "theorem q143_convergence_primitive {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (hirr : IsIrreducible P)\n (haper : IsAperiodic P) :\n IsPrimitive P := by sorry", "true_to_source": "literal", "proof": "theorem q143_convergence_primitive {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (hirr : IsIrreducible P)\n (haper : IsAperiodic P) :\n IsPrimitive P := by\n have hPnn := hP.1\n have hpow_nonneg : ∀ (k : ℕ) (i j : Ω), 0 ≤ (P ^ k) i j := by\n intro k\n induction k with\n | zero =>\n intro i j\n rw [pow_zero, Matrix.one_apply]\n split <;> norm_num\n | succ n ih =>\n intro i j\n rw [pow_succ, Matrix.mul_apply]\n apply Finset.sum_nonneg\n intro z _\n exact mul_nonneg (ih i z) (hPnn z j)\n have hmul_ge : ∀ (a b : ℕ) (x z y : Ω),\n (P ^ a) x z * (P ^ b) z y ≤ (P ^ (a + b)) x y := by\n intro a b x z y\n rw [pow_add, Matrix.mul_apply]\n refine Finset.single_le_sum (f := fun w => (P ^ a) x w * (P ^ b) w y) ?_ (Finset.mem_univ z)\n intro w _\n exact mul_nonneg (hpow_nonneg a x w) (hpow_nonneg b w y)\n have hdiag : ∀ x : Ω, ∃ Nx : ℕ, ∀ m : ℕ, Nx ≤ m → 0 < (P ^ m) x x := by\n intro x\n let M : AddSubmonoid ℕ :=\n { carrier := {n | 0 < (P ^ n) x x}\n zero_mem' := by\n simp only [Set.mem_setOf_eq, pow_zero, Matrix.one_apply_eq]; norm_num\n add_mem' := by\n intro a b ha hb\n simp only [Set.mem_setOf_eq] at *\n have h := hmul_ge a b x x x\n have hp := mul_pos ha hb\n linarith }\n have hgcd_le : Nat.setGcd (M : Set ℕ) ≤ 1 := by\n apply haper x\n intro n hn\n exact Nat.setGcd_dvd_of_mem hn\n obtain ⟨k, hk1, hkpos⟩ := hirr x x\n have hgcd_pos : 1 ≤ Nat.setGcd (M : Set ℕ) := by\n rcases Nat.eq_zero_or_pos (Nat.setGcd (M : Set ℕ)) with h0 | h1\n · have hd := Nat.setGcd_dvd_of_mem (s := (M : Set ℕ)) (show k ∈ (M : Set ℕ) from hkpos)\n rw [h0] at hd\n have hk0 : k = 0 := Nat.eq_zero_of_zero_dvd hd\n omega\n · exact h1\n have hgcd_eq : Nat.setGcd (M : Set ℕ) = 1 := le_antisymm hgcd_le hgcd_pos\n obtain ⟨Nx, hNx⟩ := Nat.exists_mem_closure_of_ge (M : Set ℕ)\n refine ⟨Nx, fun m hm => ?_⟩\n have hmem : m ∈ AddSubmonoid.closure (M : Set ℕ) :=\n hNx m hm (by rw [hgcd_eq]; exact one_dvd m)\n rw [AddSubmonoid.closure_eq M] at hmem\n exact hmem\n choose N hN using hdiag\n choose K hK using hirr\n refine ⟨Finset.univ.sup N + Finset.univ.sup (fun p : Ω × Ω => K p.1 p.2), ?_⟩\n intro x y n hn\n have hNxle : N x ≤ Finset.univ.sup N := Finset.le_sup (Finset.mem_univ x)\n have hkle : K x y ≤ Finset.univ.sup (fun p : Ω × Ω => K p.1 p.2) := by\n have h := Finset.le_sup (f := fun p : Ω × Ω => K p.1 p.2) (Finset.mem_univ (x, y))\n simpa using h\n have hdiagpos : 0 < (P ^ (n - K x y)) x x := by\n apply hN x\n omega\n have hxy : 0 < (P ^ (K x y)) x y := (hK x y).2\n have hbound := hmul_ge (n - K x y) (K x y) x x y\n have hsum : (n - K x y) + K x y = n := by omega\n rw [hsum] at hbound\n have hp := mul_pos hdiagpos hxy\n linarith", "time_seconds": 132, "topic": "Markov chains (finite & countable)" }, { "question_id": 145, "problem_name": "stochastic_processes_two_threshold_possible_values", "informal_statement": "Let (S_n) be a random walk with i.i.d. steps in {−1,0,1} and mean 0, stopped at the first trial J with S_n ≥ a or S_n ≤ b for integer thresholds a > 0 > b.\n\nShow that S_J ∈ {a, b}, i.e. the stopped walk lands exactly on a threshold with no overshoot.", "lean_statement": "theorem q145_two_threshold_possible_values (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hstepval : ∀ i ω, X i ω = -1 ∨ X i ω = 0 ∨ X i ω = 1)\n (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (S : ℕ → Ω → ℝ) (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, X i ω)\n (a b : ℤ) (ha : 0 < a) (hb : b < 0)\n (J : Ω → ℕ) (hJpos : ∀ ω, 0 < J ω)\n (hJcross : ∀ ω, (a : ℝ) ≤ S (J ω) ω ∨ S (J ω) ω ≤ (b : ℝ))\n (hJfirst : ∀ ω n, n < J ω → (b : ℝ) < S n ω ∧ S n ω < (a : ℝ)) :\n ∀ ω, S (J ω) ω = (a : ℝ) ∨ S (J ω) ω = (b : ℝ) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q145_two_threshold_possible_values (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hstepval : ∀ i ω, X i ω = -1 ∨ X i ω = 0 ∨ X i ω = 1)\n (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (S : ℕ → Ω → ℝ) (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, X i ω)\n (a b : ℤ) (ha : 0 < a) (hb : b < 0)\n (J : Ω → ℕ) (hJpos : ∀ ω, 0 < J ω)\n (hJcross : ∀ ω, (a : ℝ) ≤ S (J ω) ω ∨ S (J ω) ω ≤ (b : ℝ))\n (hJfirst : ∀ ω n, n < J ω → (b : ℝ) < S n ω ∧ S n ω < (a : ℝ)) :\n ∀ ω, S (J ω) ω = (a : ℝ) ∨ S (J ω) ω = (b : ℝ) := by\n intro ω\n have hint : ∀ n, ∃ k : ℤ, S n ω = (k : ℝ) := by\n intro n\n induction n with\n | zero => exact ⟨0, by rw [hS]; simp⟩\n | succ p ih =>\n obtain ⟨k, hk⟩ := ih\n rcases hstepval p ω with h | h | h\n · exact ⟨k - 1, by rw [hS (p+1) ω, Finset.sum_range_succ, ← hS p ω, hk, h]; push_cast; ring⟩\n · exact ⟨k, by rw [hS (p+1) ω, Finset.sum_range_succ, ← hS p ω, hk, h]; push_cast; ring⟩\n · exact ⟨k + 1, by rw [hS (p+1) ω, Finset.sum_range_succ, ← hS p ω, hk, h]; push_cast; ring⟩\n have hmpos := hJpos ω\n obtain ⟨p, hp⟩ : ∃ p, J ω = p + 1 := ⟨J ω - 1, by omega⟩\n have hsplit : S (J ω) ω = S p ω + X p ω := by\n rw [hp, hS (p+1) ω, Finset.sum_range_succ, ← hS p ω]\n have hfirst := hJfirst ω p (by omega)\n obtain ⟨k, hk⟩ := hint (J ω)\n have hstep := hstepval p ω\n rcases hJcross ω with hcross | hcross\n · left\n have hub : S (J ω) ω < (a : ℝ) + 1 := by\n rcases hstep with h | h | h <;> (rw [hsplit, h]; linarith [hfirst.2])\n rw [hk] at hcross hub\n have h1 : a ≤ k := by exact_mod_cast hcross\n have h2 : k < a + 1 := by exact_mod_cast hub\n have hkey : k = a := by omega\n rw [hk, hkey]\n · right\n have hlb : (b : ℝ) - 1 < S (J ω) ω := by\n rcases hstep with h | h | h <;> (rw [hsplit, h]; linarith [hfirst.1])\n rw [hk] at hcross hlb\n have h1 : k ≤ b := by exact_mod_cast hcross\n have h2 : b - 1 < k := by exact_mod_cast hlb\n have hkey : k = b := by omega\n rw [hk, hkey]", "time_seconds": 53, "topic": "Random walks & large deviations" }, { "question_id": 147, "problem_name": "stochastic_processes_continuity_wp1", "informal_statement": "Let (Z_n) be random variables with Z_n → α with probability 1, and let f be continuous at α.\n\nShow that f(Z_n) → f(α) with probability 1.", "lean_statement": "theorem q147_continuity_wp1 (μ : Measure Ω)\n (Z : ℕ → Ω → ℝ) (α : ℝ) (f : ℝ → ℝ) (hf : ContinuousAt f α)\n (hZ : ∀ᵐ ω ∂μ, Tendsto (fun n => Z n ω) atTop (𝓝 α)) :\n ∀ᵐ ω ∂μ, Tendsto (fun n => f (Z n ω)) atTop (𝓝 (f α)) := by\n sorry", "true_to_source": "literal", "proof": "theorem q147_continuity_wp1 (μ : Measure Ω)\n (Z : ℕ → Ω → ℝ) (α : ℝ) (f : ℝ → ℝ) (hf : ContinuousAt f α)\n (hZ : ∀ᵐ ω ∂μ, Tendsto (fun n => Z n ω) atTop (𝓝 α)) :\n ∀ᵐ ω ∂μ, Tendsto (fun n => f (Z n ω)) atTop (𝓝 (f α)) := by\n filter_upwards [hZ] with ω hω\n exact (hf.tendsto).comp hω", "time_seconds": 80, "topic": "Weak convergence & functional limits" }, { "question_id": 154, "problem_name": "stochastic_processes_column_range_monotone", "informal_statement": "Let [P] be the transition matrix of a finite-state Markov chain. For fixed j and n, let U_n = max_i P_{ij}^n and L_n = min_i P_{ij}^n.\n\nShow that U_{n+1} ≤ U_n and L_{n+1} ≥ L_n, so the range of the jth column of [P]^n is nonincreasing in n.", "lean_statement": "theorem q154_column_range_monotone {Ω : Type*} [Fintype Ω] [DecidableEq Ω] [Nonempty Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) :\n ∀ (j : Ω) (n : ℕ),\n (⨆ i, (P ^ (n + 1)) i j) ≤ (⨆ i, (P ^ n) i j) ∧\n (⨅ i, (P ^ n) i j) ≤ (⨅ i, (P ^ (n + 1)) i j) := by\n sorry", "true_to_source": "literal", "proof": "theorem q154_column_range_monotone {Ω : Type*} [Fintype Ω] [DecidableEq Ω] [Nonempty Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) :\n ∀ (j : Ω) (n : ℕ),\n (⨆ i, (P ^ (n + 1)) i j) ≤ (⨆ i, (P ^ n) i j) ∧\n (⨅ i, (P ^ n) i j) ≤ (⨅ i, (P ^ (n + 1)) i j) := by\n obtain ⟨hnn, hrow⟩ := hP\n intro j n\n have hexp : ∀ i, (P ^ (n + 1)) i j = ∑ k, P i k * (P ^ n) k j := by\n intro i; rw [pow_succ']; simp only [Matrix.mul_apply]\n constructor\n · apply ciSup_le\n intro i\n rw [hexp i]\n have hbdd : BddAbove (Set.range (fun i => (P ^ n) i j)) :=\n Set.Finite.bddAbove (Set.finite_range _)\n calc ∑ k, P i k * (P ^ n) k j\n ≤ ∑ k, P i k * (⨆ i, (P ^ n) i j) := by\n apply Finset.sum_le_sum\n intro k _\n exact mul_le_mul_of_nonneg_left (le_ciSup hbdd k) (hnn i k)\n _ = (∑ k, P i k) * (⨆ i, (P ^ n) i j) := by rw [← Finset.sum_mul]\n _ = (⨆ i, (P ^ n) i j) := by rw [hrow i, one_mul]\n · apply le_ciInf\n intro i\n rw [hexp i]\n have hbdd : BddBelow (Set.range (fun i => (P ^ n) i j)) :=\n Set.Finite.bddBelow (Set.finite_range _)\n calc ⨅ i, (P ^ n) i j\n = (∑ k, P i k) * (⨅ i, (P ^ n) i j) := by rw [hrow i, one_mul]\n _ = ∑ k, P i k * (⨅ i, (P ^ n) i j) := by rw [Finset.sum_mul]\n _ ≤ ∑ k, P i k * (P ^ n) k j := by\n apply Finset.sum_le_sum\n intro k _\n exact mul_le_mul_of_nonneg_left (ciInf_le hbdd k) (hnn i k)", "time_seconds": 76, "topic": "Markov chains (finite & countable)" }, { "question_id": 156, "problem_name": "stochastic_processes_geometric_brownian_mean", "informal_statement": "Let S(t) = S0 · exp(m t + σ B(t)) be a geometric Brownian motion, where B is standard Brownian motion (B(t) ~ Normal(0, t)). Show that its mean is E[S(t)] = S0 · exp((m + σ²/2) t).", "lean_statement": "theorem q156_geometric_brownian_mean (μ : Measure Ω) [IsProbabilityMeasure μ]\n (B : ℝ → Ω → ℝ) (S0 mdrift σ : ℝ)\n (hlaw : ∀ t : ℝ, 0 ≤ t → HasLaw (B t) (gaussianReal 0 t.toNNReal) μ)\n (S : ℝ → Ω → ℝ) (hS : ∀ t ω, S t ω = S0 * Real.exp (mdrift * t + σ * B t ω))\n (hint : ∀ t, Integrable (S t) μ) :\n ∀ t : ℝ, 0 ≤ t → ∫ ω, S t ω ∂μ = S0 * Real.exp ((mdrift + σ ^ 2 / 2) * t) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q156_geometric_brownian_mean (μ : Measure Ω) [IsProbabilityMeasure μ]\n (B : ℝ → Ω → ℝ) (S0 mdrift σ : ℝ)\n (hlaw : ∀ t : ℝ, 0 ≤ t → HasLaw (B t) (gaussianReal 0 t.toNNReal) μ)\n (S : ℝ → Ω → ℝ) (hS : ∀ t ω, S t ω = S0 * Real.exp (mdrift * t + σ * B t ω))\n (hint : ∀ t, Integrable (S t) μ) :\n ∀ t : ℝ, 0 ≤ t → ∫ ω, S t ω ∂μ = S0 * Real.exp ((mdrift + σ ^ 2 / 2) * t) := by\n intro t ht\n have hmap : μ.map (B t) = gaussianReal 0 t.toNNReal := (hlaw t ht).map_eq\n have hmgf : ∫ ω, Real.exp (σ * B t ω) ∂μ = Real.exp (t * σ ^ 2 / 2) := by\n have h := mgf_gaussianReal hmap σ\n simp only [mgf, zero_mul, zero_add, Real.coe_toNNReal t ht] at h\n exact h\n simp_rw [hS, Real.exp_add, ← mul_assoc]\n rw [integral_const_mul, hmgf, mul_assoc, ← Real.exp_add,\n show mdrift * t + t * σ ^ 2 / 2 = (mdrift + σ ^ 2 / 2) * t from by ring]", "time_seconds": 156, "topic": "Brownian motion & stochastic calculus" }, { "question_id": 157, "problem_name": "stochastic_processes_exponential_memoryless", "informal_statement": "Let X have an exponential distribution with rate r > 0. Show the memoryless property: for all s, t ≥ 0, P(X > s + t) = P(X > s) · P(X > t).", "lean_statement": "theorem q157_exponential_memoryless (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : Ω → ℝ) (r : ℝ) (hr : 0 < r) (hlaw : HasLaw X (expMeasure r) μ) :\n ∀ s t : ℝ, 0 ≤ s → 0 ≤ t →\n (μ {ω | s + t < X ω}).toReal\n = (μ {ω | s < X ω}).toReal * (μ {ω | t < X ω}).toReal := by\n sorry", "true_to_source": "abstract", "proof": "theorem q157_exponential_memoryless (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : Ω → ℝ) (r : ℝ) (hr : 0 < r) (hlaw : HasLaw X (expMeasure r) μ) :\n ∀ s t : ℝ, 0 ≤ s → 0 ≤ t →\n (μ {ω | s + t < X ω}).toReal\n = (μ {ω | s < X ω}).toReal * (μ {ω | t < X ω}).toReal := by\n haveI hp : IsProbabilityMeasure (expMeasure r) := isProbabilityMeasure_expMeasure hr\n have tail : ∀ c : ℝ, 0 ≤ c → (μ {ω | c < X ω}).toReal = Real.exp (-(r * c)) := by\n intro c hc\n have hmeas : MeasurableSet {x : ℝ | c < x} := measurableSet_Ioi\n rw [hlaw.measure_eq hmeas]\n have hset : {x : ℝ | c < x} = (Set.Iic c)ᶜ := by\n ext x; simp\n have hle : expMeasure r (Set.Iic c) ≤ 1 := by\n simpa using measure_mono (μ := expMeasure r) (Set.subset_univ (Set.Iic c))\n rw [hset, measure_compl measurableSet_Iic (measure_ne_top _ _), measure_univ,\n ENNReal.toReal_sub_of_le hle ENNReal.one_ne_top, ENNReal.toReal_one,\n ← measureReal_def, ← cdf_eq_real, cdf_expMeasure_eq hr, if_pos hc]\n ring\n intro s t hs ht\n rw [tail (s + t) (by linarith), tail s hs, tail t ht, ← Real.exp_add]\n congr 1\n ring", "time_seconds": 156, "topic": "Poisson processes" }, { "question_id": 165, "problem_name": "stochastic_processes_gg1_idle_time_identity", "informal_statement": "Consider a G/G/1 queue with queue-length process Q and cumulative busy time B(t).\n\nShow that the cumulative idle time satisfies I(t) = t − B(t) = ∫_0^t 1{Q(s) = 0} ds.", "lean_statement": "theorem q165_gg1_idle_time_identity\n (Q : ℝ → ℕ) (hQmeas : Measurable Q) (B I : ℝ → ℝ)\n (hbusy : ∀ t : ℝ, 0 ≤ t → B t = ∫ s in (0 : ℝ)..t, (if 0 < Q s then (1 : ℝ) else 0))\n (hIdef : ∀ t, I t = t - B t) :\n ∀ t : ℝ, 0 ≤ t → I t = ∫ s in (0 : ℝ)..t, (if Q s = 0 then (1 : ℝ) else 0) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q165_gg1_idle_time_identity\n (Q : ℝ → ℕ) (hQmeas : Measurable Q) (B I : ℝ → ℝ)\n (hbusy : ∀ t : ℝ, 0 ≤ t → B t = ∫ s in (0 : ℝ)..t, (if 0 < Q s then (1 : ℝ) else 0))\n (hIdef : ∀ t, I t = t - B t) :\n ∀ t : ℝ, 0 ≤ t → I t = ∫ s in (0 : ℝ)..t, (if Q s = 0 then (1 : ℝ) else 0) := by\n have hmeas : Measurable (fun s => if 0 < Q s then (1:ℝ) else 0) := by\n apply Measurable.ite _ measurable_const measurable_const\n exact measurableSet_lt measurable_const hQmeas\n intro t ht\n rw [hIdef t, hbusy t ht]\n have hint1 : IntervalIntegrable (fun s => if 0 < Q s then (1:ℝ) else 0) volume 0 t := by\n apply (intervalIntegrable_const (c := (1:ℝ))).mono_fun' hmeas.aestronglyMeasurable\n filter_upwards with s\n rw [Real.norm_eq_abs]\n split_ifs <;> norm_num\n have hpt : ∀ s, (if Q s = 0 then (1:ℝ) else 0)\n = (1:ℝ) - (if 0 < Q s then (1:ℝ) else 0) := by\n intro s\n by_cases h : Q s = 0\n · simp [h]\n · simp [h, Nat.pos_of_ne_zero h]\n have hcong : (∫ s in (0:ℝ)..t, (if Q s = 0 then (1:ℝ) else 0))\n = ∫ s in (0:ℝ)..t, ((1:ℝ) - (if 0 < Q s then (1:ℝ) else 0)) := by\n apply intervalIntegral.integral_congr\n intro s _; exact hpt s\n rw [hcong, intervalIntegral.integral_sub intervalIntegrable_const hint1,\n intervalIntegral.integral_const]\n simp", "time_seconds": 54, "topic": "Continuous-time Markov & queues" }, { "question_id": 168, "problem_name": "stochastic_processes_scaled_branching_martingale", "informal_statement": "Consider a branching process (X_n) with i.i.d. offspring counts of mean Ȳ, and let Z_n = X_n / Ȳ^n.\n\nShow that (Z_n) is a martingale.", "lean_statement": "theorem q168_branching_martingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hadap : Adapted ℱ X) (hint : ∀ n, Integrable (X n) μ)\n (Ybar : ℝ) (hY : 0 < Ybar)\n (hbranch : ∀ n, μ[X (n + 1) | ℱ n] =ᵐ[μ] fun ω => Ybar * X n ω) :\n Martingale (fun n ω => X n ω / Ybar ^ n) ℱ μ := by\n sorry", "true_to_source": "abstract", "proof": "theorem q168_branching_martingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hadap : Adapted ℱ X) (hint : ∀ n, Integrable (X n) μ)\n (Ybar : ℝ) (hY : 0 < Ybar)\n (hbranch : ∀ n, μ[X (n + 1) | ℱ n] =ᵐ[μ] fun ω => Ybar * X n ω) :\n Martingale (fun n ω => X n ω / Ybar ^ n) ℱ μ := by\n have hYne : Ybar ≠ 0 := hY.ne'\n refine martingale_nat ?_ ?_ ?_\n · intro n\n have h := hadap.stronglyAdapted n\n simp only [div_eq_mul_inv]\n exact h.mul stronglyMeasurable_const\n · intro n\n exact (hint n).div_const (Ybar ^ n)\n · intro i\n have hstep1 : (fun ω => X (i + 1) ω / Ybar ^ (i + 1))\n = (Ybar ^ (i + 1))⁻¹ • X (i + 1) := by\n funext ω; simp [div_eq_mul_inv, smul_eq_mul, mul_comm]\n have hce : μ[(fun ω => X (i + 1) ω / Ybar ^ (i + 1)) | ℱ i]\n =ᵐ[μ] (Ybar ^ (i + 1))⁻¹ • μ[X (i + 1) | ℱ i] := by\n rw [hstep1]; exact condExp_smul _ _ _\n have hfin : ((Ybar ^ (i + 1))⁻¹ • μ[X (i + 1) | ℱ i])\n =ᵐ[μ] fun ω => X i ω / Ybar ^ i := by\n filter_upwards [hbranch i] with ω hω\n simp only [Pi.smul_apply, smul_eq_mul, hω]\n rw [pow_succ]\n field_simp\n exact (hce.trans hfin).symm", "time_seconds": 207, "topic": "Martingales & stopping" }, { "question_id": 169, "problem_name": "stochastic_processes_gamblers_ruin_2", "informal_statement": "Let X = (X_n)_{n ≥ 0} be Gambler’s ruin with state space Ω = {0, 1, 2, ..., N}:\n X_0 = k, ℙ[X_{n + 1} = X_n + 1 | X_n] = 1/2, τ = min{n: X_n = 0 or N}.\n You are given the martingale Y = (Yn := X_n^2 − n)_{n ≥ 0}.\n\n Show that Y has bounded increments.", "lean_statement": "theorem q169_gambler_bounded_increments (μ : Measure Ω) [IsProbabilityMeasure μ] (N : ℕ)\n (X : ℕ → Ω → ℝ) (hXbd : ∀ n ω, 0 ≤ X n ω ∧ X n ω ≤ N)\n (hstep1 : ∀ n, ∀ᵐ ω ∂μ, |X (n + 1) ω - X n ω| = 1) :\n ∃ M : ℝ, ∀ n, ∀ᵐ ω ∂μ,\n |((X (n + 1) ω) ^ 2 - (n + 1)) - ((X n ω) ^ 2 - n)| ≤ M := by sorry", "true_to_source": "literal", "proof": "theorem q169_gambler_bounded_increments (μ : Measure Ω) [IsProbabilityMeasure μ] (N : ℕ)\n (X : ℕ → Ω → ℝ) (hXbd : ∀ n ω, 0 ≤ X n ω ∧ X n ω ≤ N)\n (hstep1 : ∀ n, ∀ᵐ ω ∂μ, |X (n + 1) ω - X n ω| = 1) :\n ∃ M : ℝ, ∀ n, ∀ᵐ ω ∂μ,\n |((X (n + 1) ω) ^ 2 - (n + 1)) - ((X n ω) ^ 2 - n)| ≤ M := by\n refine ⟨2 * N + 1, ?_⟩\n intro n\n filter_upwards [hstep1 n] with ω hω\n obtain ⟨ha0, haN⟩ := hXbd n ω\n obtain ⟨hb0, hbN⟩ := hXbd (n + 1) ω\n rcases (abs_eq (by norm_num : (0:ℝ) ≤ 1)).mp hω with h | h\n · rw [abs_le]; push_cast; constructor <;> nlinarith\n · rw [abs_le]; push_cast; constructor <;> nlinarith", "time_seconds": 67, "topic": "Martingales & stopping" }, { "question_id": 177, "problem_name": "stochastic_processes_supermartingale_convergence", "informal_statement": "Let X = (X_n)_{n ≥ 0} be a non-negative supermartingale.\n\n Show that X_n converges a.s. to some a.s. finite limit.", "lean_statement": "theorem q177_super_convergence (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hX : Supermartingale X ℱ μ) (hnn : ∀ n ω, 0 ≤ X n ω) :\n ∃ Xinf : Ω → ℝ, ∀ᵐ ω ∂μ, Tendsto (fun n => X n ω) atTop (𝓝 (Xinf ω)) := by sorry", "true_to_source": "literal", "proof": "theorem q177_super_convergence (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hX : Supermartingale X ℱ μ) (hnn : ∀ n ω, 0 ≤ X n ω) :\n ∃ Xinf : Ω → ℝ, ∀ᵐ ω ∂μ, Tendsto (fun n => X n ω) atTop (𝓝 (Xinf ω)) := by\n have hsub : Submartingale (-X) ℱ μ := hX.neg\n have hint_le : ∀ n, ∫ ω, X n ω ∂μ ≤ ∫ ω, X 0 ω ∂μ := by\n intro n\n have h := hX.setIntegral_le (Nat.zero_le n) (s := Set.univ) MeasurableSet.univ\n simpa [setIntegral_univ] using h\n have hbdd : ∀ n, eLpNorm ((-X) n) 1 μ ≤ ENNReal.ofReal (∫ ω, X 0 ω ∂μ) := by\n intro n\n have hne : ((-X) n) = -(X n) := rfl\n rw [hne, eLpNorm_neg, eLpNorm_one_eq_lintegral_enorm]\n have hnn' : 0 ≤ᵐ[μ] X n := ae_of_all _ (hnn n)\n have hInt : Integrable (X n) μ := hX.integrable n\n have heq : ∫⁻ ω, ‖X n ω‖ₑ ∂μ = ENNReal.ofReal (∫ ω, X n ω ∂μ) := by\n rw [ofReal_integral_eq_lintegral_ofReal hInt hnn']\n apply lintegral_congr_ae\n filter_upwards [hnn'] with ω hω\n rw [Real.enorm_eq_ofReal hω]\n rw [heq]\n exact ENNReal.ofReal_le_ofReal (hint_le n)\n refine ⟨fun ω => -(ℱ.limitProcess (-X) μ ω), ?_⟩\n filter_upwards [hsub.ae_tendsto_limitProcess hbdd] with ω hω\n simpa only [Pi.neg_apply, neg_neg] using hω.neg", "time_seconds": 273, "topic": "Martingales & stopping" }, { "question_id": 185, "problem_name": "stochastic_processes_stopped_martingale_preserves", "informal_statement": "Let (Z_n) be a martingale and let J be a stopping trial; let Z_n* be the stopped process (frozen at value Z_J for n ≥ J).\n\nShow that (Z_n*) is also a martingale.", "lean_statement": "theorem q185_stopped_martingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (Z : ℕ → Ω → ℝ) (hZ : Martingale Z ℱ μ) (J : Ω → ℕ)\n (hJ : IsStoppingTime ℱ (natStop J)) :\n Martingale (stoppedProcess Z (natStop J)) ℱ μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q185_stopped_martingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (Z : ℕ → Ω → ℝ) (hZ : Martingale Z ℱ μ) (J : Ω → ℕ)\n (hJ : IsStoppingTime ℱ (natStop J)) :\n Martingale (stoppedProcess Z (natStop J)) ℱ μ := by\n have hneg : stoppedProcess (-Z) (natStop J) = -stoppedProcess Z (natStop J) := by\n funext i ω\n simp only [stoppedProcess, Pi.neg_apply]\n rw [martingale_iff]\n refine ⟨?_, hZ.submartingale.stoppedProcess hJ⟩\n have hsub_neg : Submartingale (stoppedProcess (-Z) (natStop J)) ℱ μ :=\n (hZ.neg.submartingale).stoppedProcess hJ\n rw [hneg] at hsub_neg\n simpa using hsub_neg.neg", "time_seconds": 89, "topic": "Martingales & stopping" }, { "question_id": 187, "problem_name": "stochastic_processes_mm1_expected_delay", "informal_statement": "Consider an M/M/1 queue in steady state with arrival rate λ, service rate μ, and ρ = λ/μ < 1.\n\nShow that the expected time a customer spends in the system is 1/(μ − λ).", "lean_statement": "theorem q187_mm1_expected_delay\n (lam mu : ℝ) (hlam : 0 < lam) (hmu : 0 < mu)\n (ρ : ℝ) (hρ : ρ = lam / mu) (hρ1 : ρ < 1)\n (p : ℕ → ℝ) (hp0 : ∀ i, 0 ≤ p i) (hpsum : ∑' i, p i = 1)\n (hbal0 : p 0 * lam = p 1 * mu)\n (hbal : ∀ n, p (n + 1) * (lam + mu) = p n * lam + p (n + 2) * mu)\n (L : ℝ) (hL : L = ∑' i : ℕ, (i : ℝ) * p i)\n (W : ℝ) (hlittle : L = lam * W) :\n W = 1 / (mu - lam) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q187_mm1_expected_delay\n (lam mu : ℝ) (hlam : 0 < lam) (hmu : 0 < mu)\n (ρ : ℝ) (hρ : ρ = lam / mu) (hρ1 : ρ < 1)\n (p : ℕ → ℝ) (hp0 : ∀ i, 0 ≤ p i) (hpsum : ∑' i, p i = 1)\n (hbal0 : p 0 * lam = p 1 * mu)\n (hbal : ∀ n, p (n + 1) * (lam + mu) = p n * lam + p (n + 2) * mu)\n (L : ℝ) (hL : L = ∑' i : ℕ, (i : ℝ) * p i)\n (W : ℝ) (hlittle : L = lam * W) :\n W = 1 / (mu - lam) := by\n have hmune : mu ≠ 0 := ne_of_gt hmu\n have hρpos : 0 < ρ := by rw [hρ]; exact div_pos hlam hmu\n have hρnonneg : 0 ≤ ρ := le_of_lt hρpos\n have h1mρ : (1 : ℝ) - ρ ≠ 0 := by\n have : 0 < 1 - ρ := by linarith\n exact ne_of_gt this\n have hlm : lam < mu := by rw [hρ] at hρ1; exact (div_lt_one hmu).mp hρ1\n have hmlam : 0 < mu - lam := by linarith\n have hmlamne : mu - lam ≠ 0 := ne_of_gt hmlam\n -- local balance\n have hlocal : ∀ n, p (n + 1) * mu = p n * lam := by\n intro n\n induction n with\n | zero => linarith [hbal0]\n | succ k ih =>\n have hb := hbal k\n rw [mul_add] at hb\n linarith [ih, hb]\n have hgeo : ∀ n, p n = ρ ^ n * p 0 := by\n intro n\n induction n with\n | zero => simp\n | succ k ih =>\n have hstep : p (k + 1) = p k * ρ := by\n have h := hlocal k\n rw [hρ]; field_simp; linarith [h]\n rw [hstep, ih, pow_succ]; ring\n have hcompute : ∑' i, p i = (1 - ρ)⁻¹ * p 0 := by\n rw [tsum_congr hgeo, tsum_mul_right, tsum_geometric_of_lt_one hρnonneg hρ1]\n have hsum2 : (1 - ρ)⁻¹ * p 0 = 1 := hcompute ▸ hpsum\n have hp0val : p 0 = 1 - ρ := by\n field_simp at hsum2; linarith [hsum2]\n have hpart1 : ∀ i, p i = (1 - ρ) * ρ ^ i := by\n intro i; rw [hgeo i, hp0val]; ring\n have hnorm : ‖ρ‖ < 1 := by\n rw [Real.norm_eq_abs, abs_of_nonneg hρnonneg]; exact hρ1\n have hLcomp : ∑' i : ℕ, (i : ℝ) * p i = ρ / (1 - ρ) := by\n calc ∑' i : ℕ, (i : ℝ) * p i\n = ∑' i : ℕ, (1 - ρ) * ((i : ℝ) * ρ ^ i) := by\n apply tsum_congr; intro i; rw [hpart1 i]; ring\n _ = (1 - ρ) * ∑' i : ℕ, (i : ℝ) * ρ ^ i := tsum_mul_left\n _ = (1 - ρ) * (ρ / (1 - ρ) ^ 2) := by\n rw [tsum_coe_mul_geometric_of_norm_lt_one hnorm]\n _ = ρ / (1 - ρ) := by field_simp\n -- combine\n have key : lam * W = lam * (1 / (mu - lam)) := by\n rw [← hlittle, hL, hLcomp, hρ]\n field_simp\n exact mul_left_cancel₀ (ne_of_gt hlam) key", "time_seconds": 242, "topic": "Continuous-time Markov & queues" }, { "question_id": 188, "problem_name": "stochastic_processes_accessibility_transitive", "informal_statement": "Let (X_n) be a finite-state Markov chain, and say i → j if j is accessible from i.\n\nShow that accessibility is transitive: if i → j and j → k then i → k.", "lean_statement": "theorem q188_accessibility_transitive {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (i j k : Ω)\n (hij : ∃ m : ℕ, 0 < (P ^ m) i j) (hjk : ∃ n : ℕ, 0 < (P ^ n) j k) :\n ∃ p : ℕ, 0 < (P ^ p) i k := by\n sorry", "true_to_source": "literal", "proof": "theorem q188_accessibility_transitive {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (i j k : Ω)\n (hij : ∃ m : ℕ, 0 < (P ^ m) i j) (hjk : ∃ n : ℕ, 0 < (P ^ n) j k) :\n ∃ p : ℕ, 0 < (P ^ p) i k := by\n obtain ⟨m, hm⟩ := hij\n obtain ⟨n, hn⟩ := hjk\n -- nonnegativity of powers\n have hnn : ∀ (t : ℕ) (x y : Ω), 0 ≤ (P ^ t) x y := by\n intro t\n induction t with\n | zero => intro x y; simp [Matrix.one_apply]; split <;> norm_num\n | succ s ih =>\n intro x y\n rw [pow_succ, Matrix.mul_apply]\n exact Finset.sum_nonneg (fun l _ => mul_nonneg (ih x l) (hP.1 l y))\n refine ⟨m + n, ?_⟩\n rw [pow_add, Matrix.mul_apply]\n have hterm : 0 < (P ^ m) i j * (P ^ n) j k := mul_pos hm hn\n calc 0 < (P ^ m) i j * (P ^ n) j k := hterm\n _ ≤ ∑ l, (P ^ m) i l * (P ^ n) l k :=\n Finset.single_le_sum (f := fun l => (P ^ m) i l * (P ^ n) l k)\n (fun l _ => mul_nonneg (hnn m i l) (hnn n l k)) (Finset.mem_univ j)", "time_seconds": 65, "topic": "Markov chains (finite & countable)" }, { "question_id": 189, "problem_name": "stochastic_processes_transitivity", "informal_statement": "For a transitive Markov chain on finite state space Ω, the uniform measure is stationary.", "lean_statement": "theorem q189_transitivity {Ω : Type*} [Fintype Ω] [Nonempty Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P)\n (htrans : ∀ x y : Ω, ∃ φ : Ω ≃ Ω, φ x = y ∧ ∀ a b, P (φ a) (φ b) = P a b) :\n IsStationary P (fun _ => 1 / (Fintype.card Ω : ℝ)) := by sorry", "true_to_source": "literal", "proof": "theorem q189_transitivity {Ω : Type*} [Fintype Ω] [Nonempty Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P)\n (htrans : ∀ x y : Ω, ∃ φ : Ω ≃ Ω, φ x = y ∧ ∀ a b, P (φ a) (φ b) = P a b) :\n IsStationary P (fun _ => 1 / (Fintype.card Ω : ℝ)) := by\n have hN : (Fintype.card Ω : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero\n have hcol : ∀ y y' : Ω, ∑ x, P x y = ∑ x, P x y' := by\n intro y y'\n symm\n obtain ⟨φ, hφ, hPφ⟩ := htrans y y'\n calc ∑ x, P x y' = ∑ x, P x (φ y) := by rw [hφ]\n _ = ∑ x, P (φ (φ.symm x)) (φ y) := by\n apply Finset.sum_congr rfl; intro x _; rw [Equiv.apply_symm_apply]\n _ = ∑ x, P (φ.symm x) y := by\n apply Finset.sum_congr rfl; intro x _; rw [hPφ]\n _ = ∑ x, P x y := Equiv.sum_comp φ.symm (fun x => P x y)\n have hcolone : ∀ y : Ω, ∑ x, P x y = 1 := by\n intro y\n have h1 : ∑ y', ∑ x, P x y' = (Fintype.card Ω : ℝ) := by\n rw [Finset.sum_comm]\n simp only [hP.2]\n rw [Finset.sum_const, Finset.card_univ]; simp\n have h2 : ∑ y' : Ω, ∑ x, P x y' = (Fintype.card Ω : ℝ) * (∑ x, P x y) := by\n rw [Finset.sum_congr rfl (fun y' _ => hcol y' y)]\n rw [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]\n have htot : (Fintype.card Ω : ℝ) * (∑ x, P x y) = (Fintype.card Ω : ℝ) := by\n rw [← h2]; exact h1\n have h3 : (Fintype.card Ω : ℝ) * (∑ x, P x y) = (Fintype.card Ω : ℝ) * 1 := by\n rw [htot, mul_one]\n exact mul_left_cancel₀ hN h3\n refine ⟨fun _ => by positivity, ?_, ?_⟩\n · rw [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]; field_simp\n · intro y\n simp only\n rw [← Finset.mul_sum, hcolone y, mul_one]", "time_seconds": 160, "topic": "Markov chains (finite & countable)" }, { "question_id": 191, "problem_name": "stochastic_processes_doob_martingale_convergence", "informal_statement": "Let (X_n) be a supermartingale with sup_n E|X_n| < ∞.\n\nShow that X_∞ = lim_{n→∞} X_n exists almost surely and satisfies E|X_∞| < ∞.", "lean_statement": "theorem q191_doob_martingale_convergence (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (X : ℕ → Ω → ℝ) (hX : Supermartingale X ℱ μ)\n (hbdd : ∃ C : ℝ, ∀ n, ∫ ω, |X n ω| ∂μ ≤ C) :\n ∃ Xinf : Ω → ℝ,\n (∀ᵐ ω ∂μ, Tendsto (fun n => X n ω) atTop (𝓝 (Xinf ω))) ∧ Integrable Xinf μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q191_doob_martingale_convergence (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (X : ℕ → Ω → ℝ) (hX : Supermartingale X ℱ μ)\n (hbdd : ∃ C : ℝ, ∀ n, ∫ ω, |X n ω| ∂μ ≤ C) :\n ∃ Xinf : Ω → ℝ,\n (∀ᵐ ω ∂μ, Tendsto (fun n => X n ω) atTop (𝓝 (Xinf ω))) ∧ Integrable Xinf μ := by\n obtain ⟨C, hC⟩ := hbdd\n have hsub : Submartingale (-X) ℱ μ := hX.neg\n have hbdd' : ∀ n, eLpNorm ((-X) n) 1 μ ≤ (Real.toNNReal C : ℝ≥0∞) := by\n intro n\n have hInt : Integrable (X n) μ := hX.integrable n\n have step : eLpNorm ((-X) n) 1 μ = ENNReal.ofReal (∫ ω, |X n ω| ∂μ) := by\n rw [show ((-X) n) = -(X n) from rfl, eLpNorm_neg, eLpNorm_one_eq_lintegral_enorm,\n ← ofReal_integral_norm_eq_lintegral_enorm hInt]\n simp only [Real.norm_eq_abs]\n rw [step]\n calc ENNReal.ofReal (∫ ω, |X n ω| ∂μ)\n ≤ ENNReal.ofReal C := ENNReal.ofReal_le_ofReal (hC n)\n _ = (Real.toNNReal C : ℝ≥0∞) := rfl\n have hae := hsub.ae_tendsto_limitProcess hbdd'\n have hmem := hsub.memLp_limitProcess (p := 1) hbdd'\n refine ⟨-(ℱ.limitProcess (-X) μ), ?_, (hmem.integrable le_rfl).neg⟩\n filter_upwards [hae] with ω hω\n have hneg := hω.neg\n simpa only [Pi.neg_apply, neg_neg] using hneg", "time_seconds": 207, "topic": "Martingales & stopping" }, { "question_id": 192, "problem_name": "stochastic_processes_brownian_exponential_martingale", "informal_statement": "For standard Brownian motion B and a real parameter θ, the exponential process exp(θ B(t) − θ² t / 2) is a martingale with respect to the natural filtration.", "lean_statement": "theorem q192_brownian_exponential_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℝ m0) (B : ℝ → Ω → ℝ) (θ : ℝ)\n (hadap : Adapted ℱ B)\n (hint : ∀ t, Integrable (fun ω => Real.exp (θ * B t ω - θ ^ 2 * t / 2)) μ)\n (hincr : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n HasLaw (fun ω => B t ω - B s ω) (gaussianReal 0 (t - s).toNNReal) μ)\n (hpast : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n Indep (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (ℱ s) μ) :\n ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n μ[(fun ω => Real.exp (θ * B t ω - θ ^ 2 * t / 2)) | (ℱ s)]\n =ᵐ[μ] (fun ω => Real.exp (θ * B s ω - θ ^ 2 * s / 2)) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q192_brownian_exponential_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℝ m0) (B : ℝ → Ω → ℝ) (θ : ℝ)\n (hadap : Adapted ℱ B)\n (hint : ∀ t, Integrable (fun ω => Real.exp (θ * B t ω - θ ^ 2 * t / 2)) μ)\n (hincr : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n HasLaw (fun ω => B t ω - B s ω) (gaussianReal 0 (t - s).toNNReal) μ)\n (hpast : ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n Indep (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (ℱ s) μ) :\n ∀ s t : ℝ, 0 ≤ s → s ≤ t →\n μ[(fun ω => Real.exp (θ * B t ω - θ ^ 2 * t / 2)) | (ℱ s)]\n =ᵐ[μ] (fun ω => Real.exp (θ * B s ω - θ ^ 2 * s / 2)) := by\n intro s t hs hst\n have hst' : (0 : ℝ) ≤ t - s := by linarith\n -- Measurability of B t, B s and the increment\n have hBt : Measurable (B t) := (hadap t).mono (ℱ.le t) le_rfl\n have hBs : Measurable (B s) := (hadap s).mono (ℱ.le s) le_rfl\n have hX_meas : Measurable (fun ω => B t ω - B s ω) := hBt.sub hBs\n have hm1le : MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance ≤ m0 :=\n measurable_iff_comap_le.mp hX_meas\n -- SigmaFinite trim (from finiteness of μ)\n haveI hsf : SigmaFinite (μ.trim (ℱ.le s)) := by\n haveI : IsFiniteMeasure (μ.trim (ℱ.le s)) := isFiniteMeasure_trim (ℱ.le s)\n infer_instance\n -- Law of the increment\n have hlaw := hincr s t hs hst\n have hmap : μ.map (fun ω => B t ω - B s ω) = gaussianReal 0 (t - s).toNNReal := hlaw.map_eq\n have haem : AEMeasurable (fun ω => B t ω - B s ω) μ := hlaw.aemeasurable\n -- Integrability of exp(θ*(B t - B s))\n have h2 : Integrable (fun ω => Real.exp (θ * (B t ω - B s ω))) μ := by\n have haesm : AEStronglyMeasurable (fun x : ℝ => Real.exp (θ * x))\n (μ.map (fun ω => B t ω - B s ω)) := by\n apply Continuous.aestronglyMeasurable; fun_prop\n have h1' : Integrable (fun x : ℝ => Real.exp (θ * x)) (μ.map (fun ω => B t ω - B s ω)) := by\n rw [hmap]; exact integrable_exp_mul_gaussianReal θ\n exact (integrable_map_measure haesm haem).mp h1'\n -- Integrability of the increment factor g\n have hg_int : Integrable (fun ω => Real.exp (θ * (B t ω - B s ω) - θ ^ 2 * (t - s) / 2)) μ := by\n have hrw : (fun ω => Real.exp (θ * (B t ω - B s ω) - θ ^ 2 * (t - s) / 2))\n = fun ω => Real.exp (-(θ ^ 2 * (t - s) / 2)) * Real.exp (θ * (B t ω - B s ω)) := by\n funext ω; rw [← Real.exp_add]; congr 1; ring\n rw [hrw]; exact h2.const_mul _\n -- Strong measurability of the ℱ s-measurable factor f\n have hf_meas : StronglyMeasurable[ℱ s] (fun ω => Real.exp (θ * B s ω - θ ^ 2 * s / 2)) :=\n Real.continuous_exp.comp_stronglyMeasurable\n (((hadap s).const_mul θ).sub measurable_const).stronglyMeasurable\n -- Strong measurability of g w.r.t. the comap σ-algebra\n have hg_m1 : StronglyMeasurable[MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance]\n (fun ω => Real.exp (θ * (B t ω - B s ω) - θ ^ 2 * (t - s) / 2)) := by\n have hXm1 : Measurable[MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance]\n (fun ω => B t ω - B s ω) := measurable_iff_comap_le.mpr le_rfl\n have hφ : Measurable (fun x : ℝ => Real.exp (θ * x - θ ^ 2 * (t - s) / 2)) := by fun_prop\n exact (hφ.comp hXm1).stronglyMeasurable\n -- f * g = the original integrand, and its integrability\n have hfg_eq : (fun ω => Real.exp (θ * B s ω - θ ^ 2 * s / 2))\n * (fun ω => Real.exp (θ * (B t ω - B s ω) - θ ^ 2 * (t - s) / 2))\n = fun ω => Real.exp (θ * B t ω - θ ^ 2 * t / 2) := by\n funext ω; simp only [Pi.mul_apply]; rw [← Real.exp_add]; congr 1; ring\n have hfg_int : Integrable ((fun ω => Real.exp (θ * B s ω - θ ^ 2 * s / 2))\n * (fun ω => Real.exp (θ * (B t ω - B s ω) - θ ^ 2 * (t - s) / 2))) μ := by\n rw [hfg_eq]; exact hint t\n -- The increment factor has mean 1 (Gaussian MGF)\n have hmgf : mgf (fun ω => B t ω - B s ω) μ θ\n = Real.exp (0 * θ + ((t - s).toNNReal : ℝ) * θ ^ 2 / 2) := mgf_gaussianReal hmap θ\n have hEg : (∫ ω, Real.exp (θ * (B t ω - B s ω) - θ ^ 2 * (t - s) / 2) ∂μ) = 1 := by\n have hrw : (fun ω => Real.exp (θ * (B t ω - B s ω) - θ ^ 2 * (t - s) / 2))\n = fun ω => Real.exp (-(θ ^ 2 * (t - s) / 2)) * Real.exp (θ * (B t ω - B s ω)) := by\n funext ω; rw [← Real.exp_add]; congr 1; ring\n rw [hrw, integral_const_mul]\n have hint_eq : (∫ ω, Real.exp (θ * (B t ω - B s ω)) ∂μ)\n = mgf (fun ω => B t ω - B s ω) μ θ := rfl\n rw [hint_eq, hmgf, Real.coe_toNNReal (t - s) hst', ← Real.exp_add]\n rw [show -(θ ^ 2 * (t - s) / 2) + (0 * θ + (t - s) * θ ^ 2 / 2) = 0 by ring, Real.exp_zero]\n -- Conditional expectation of g given ℱ s is (a.e.) the constant 1\n have hval : μ[(fun ω => Real.exp (θ * (B t ω - B s ω) - θ ^ 2 * (t - s) / 2)) | ℱ s]\n =ᵐ[μ] fun _ => (1 : ℝ) := by\n refine (condExp_indep_eq hm1le (ℱ.le s) hg_m1 (hpast s t hs hst)).trans ?_\n filter_upwards with ω\n exact hEg\n -- Assemble\n rw [show (fun ω => Real.exp (θ * B t ω - θ ^ 2 * t / 2))\n = (fun ω => Real.exp (θ * B s ω - θ ^ 2 * s / 2))\n * (fun ω => Real.exp (θ * (B t ω - B s ω) - θ ^ 2 * (t - s) / 2)) from hfg_eq.symm]\n refine (condExp_mul_of_stronglyMeasurable_left hf_meas hfg_int hg_int).trans ?_\n filter_upwards [hval] with ω hω\n simp only [Pi.mul_apply, hω, mul_one]", "time_seconds": 156, "topic": "Brownian motion & stochastic calculus" }, { "question_id": 193, "problem_name": "stochastic_processes_sprt_generating_function_root", "informal_statement": "Consider a sequential probability ratio test whose per-observation log-likelihood ratio under hypothesis 1 has semi-invariant generating function γ_1(r) = ln ∫ f_{Y|H}(y|1)^{1−r} f_{Y|H}(y|0)^r dy.\n\nShow that γ_1(1) = 0, so that r* = 1 is the relevant root for applying Wald's identity to the log-likelihood-ratio walk.", "lean_statement": "theorem q193_sprt_generating_function_root {α : Type*} [MeasurableSpace α] (ν : Measure α)\n (f0 f1 : α → ℝ) (hf0 : ∫ y, f0 y ∂ν = 1)\n (γ₁ : ℝ → ℝ)\n (hγ : ∀ r, γ₁ r = Real.log (∫ y, (f1 y) ^ (1 - r) * (f0 y) ^ r ∂ν)) :\n γ₁ 1 = 0 := by\n sorry", "true_to_source": "literal", "proof": "theorem q193_sprt_generating_function_root {α : Type*} [MeasurableSpace α] (ν : Measure α)\n (f0 f1 : α → ℝ) (hf0 : ∫ y, f0 y ∂ν = 1)\n (γ₁ : ℝ → ℝ)\n (hγ : ∀ r, γ₁ r = Real.log (∫ y, (f1 y) ^ (1 - r) * (f0 y) ^ r ∂ν)) :\n γ₁ 1 = 0 := by\n rw [hγ]\n have : (∫ y, (f1 y) ^ (1 - (1:ℝ)) * (f0 y) ^ (1:ℝ) ∂ν) = 1 := by\n simp only [sub_self, Real.rpow_zero, Real.rpow_one, one_mul]\n exact hf0\n rw [this, Real.log_one]", "time_seconds": 44, "topic": "Random walks & large deviations" }, { "question_id": 194, "problem_name": "stochastic_processes_stopping_time_indicator_predictable", "informal_statement": "Let τ be a stopping time with respect to a filtration (F_n).\n\nShow that H_n = 1{τ ≥ n} is predictable.", "lean_statement": "theorem q194_stopping_time_indicator_predictable\n (ℱ : Filtration ℕ m0) (T : Ω → ℕ) (hT : IsStoppingTime ℱ (natStop T)) :\n IsStronglyPredictable ℱ\n (fun n ω => Set.indicator {ω | n ≤ T ω} (fun _ => (1 : ℝ)) ω) := by\n sorry", "true_to_source": "literal", "proof": "theorem q194_stopping_time_indicator_predictable\n (ℱ : Filtration ℕ m0) (T : Ω → ℕ) (hT : IsStoppingTime ℱ (natStop T)) :\n IsStronglyPredictable ℱ\n (fun n ω => Set.indicator {ω | n ≤ T ω} (fun _ => (1 : ℝ)) ω) := by\n refine IsStronglyPredictable.of_measurable_add_one ?_ ?_\n · have hset : {ω : Ω | (0 : ℕ) ≤ T ω} = Set.univ := by ext ω; simp\n simp only [hset, Set.indicator_univ]\n exact stronglyMeasurable_const\n · intro n\n have hmeas : MeasurableSet[ℱ n] {ω | n + 1 ≤ T ω} := by\n have hset : {ω | n + 1 ≤ T ω} = {ω | natStop T ω ≤ (n : WithTop ℕ)}ᶜ := by\n ext ω\n simp only [natStop, Set.mem_setOf_eq, Set.mem_compl_iff, not_le]\n constructor\n · intro h; exact_mod_cast Nat.lt_of_succ_le h\n · intro h; exact_mod_cast Nat.succ_le_of_lt (by exact_mod_cast h)\n rw [hset]\n exact (hT.measurableSet_le n).compl\n exact (stronglyMeasurable_const).indicator hmeas", "time_seconds": 560, "topic": "Martingales & stopping" }, { "question_id": 195, "problem_name": "stochastic_processes_no_arbitrage_riskless", "informal_statement": "Consider a market with a bond of rate r and a risky asset with volatility σ = 0.\n\nShow that the absence of arbitrage forces the drift to satisfy μ = r.", "lean_statement": "theorem q195_no_arbitrage_riskless\n (r μ S₀ : ℝ) (hS₀ : 0 < S₀)\n (hnoarb : ∀ t : ℝ, Real.exp (-(r * t)) * (S₀ * Real.exp (μ * t)) = S₀) :\n μ = r := by\n sorry", "true_to_source": "abstract", "proof": "theorem q195_no_arbitrage_riskless\n (r μ S₀ : ℝ) (hS₀ : 0 < S₀)\n (hnoarb : ∀ t : ℝ, Real.exp (-(r * t)) * (S₀ * Real.exp (μ * t)) = S₀) :\n μ = r := by\n have h := hnoarb 1\n simp only [mul_one] at h\n have key : Real.exp (-r) * Real.exp μ = 1 :=\n mul_left_cancel₀ (ne_of_gt hS₀) (by rw [mul_one]; linear_combination h)\n rw [← Real.exp_add] at key\n have := (Real.exp_eq_one_iff _).mp key\n linarith", "time_seconds": 84, "topic": "Brownian motion & stochastic calculus" }, { "question_id": 200, "problem_name": "stochastic_processes_gg1_workload_representation", "informal_statement": "Consider a G/G/1 queue with arrival process A, service process S, cumulative busy time B(t), and workload Z(t). Let X(t) = V(Q(0) + A(t)) − t and I(t) = t − B(t).\n\nShow that Z(t) = X(t) + I(t) with Z(t) ≥ 0, that I is non-decreasing with I(0)=0, and that ∫_0^∞ Z(t) dI(t) = 0.", "lean_statement": "theorem q200_gg1_workload_representation\n (W B X I Z : ℝ → ℝ)\n (hXdef : ∀ t, X t = W t - t)\n (hIdef : ∀ t, I t = t - B t)\n (hZdef : ∀ t, Z t = W t - B t)\n (hW0 : 0 ≤ W 0) (hB0 : B 0 = 0)\n (hrate : ∀ s t : ℝ, s ≤ t → B t - B s ≤ t - s)\n (hcap : ∀ t, B t ≤ W t)\n (hbusy : ∀ s t : ℝ, s ≤ t → (∀ u ∈ Set.Icc s t, 0 < Z u) → B t - B s = t - s) :\n (∀ t, Z t = X t + I t) ∧\n (∀ t, 0 ≤ Z t) ∧\n (∀ s t : ℝ, s ≤ t → I s ≤ I t) ∧\n I 0 = 0 ∧\n (∀ s t : ℝ, s ≤ t → (∀ u ∈ Set.Icc s t, 0 < Z u) → I s = I t) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q200_gg1_workload_representation\n (W B X I Z : ℝ → ℝ)\n (hXdef : ∀ t, X t = W t - t)\n (hIdef : ∀ t, I t = t - B t)\n (hZdef : ∀ t, Z t = W t - B t)\n (hW0 : 0 ≤ W 0) (hB0 : B 0 = 0)\n (hrate : ∀ s t : ℝ, s ≤ t → B t - B s ≤ t - s)\n (hcap : ∀ t, B t ≤ W t)\n (hbusy : ∀ s t : ℝ, s ≤ t → (∀ u ∈ Set.Icc s t, 0 < Z u) → B t - B s = t - s) :\n (∀ t, Z t = X t + I t) ∧\n (∀ t, 0 ≤ Z t) ∧\n (∀ s t : ℝ, s ≤ t → I s ≤ I t) ∧\n I 0 = 0 ∧\n (∀ s t : ℝ, s ≤ t → (∀ u ∈ Set.Icc s t, 0 < Z u) → I s = I t) := by\n refine ⟨?_, ?_, ?_, ?_, ?_⟩\n · intro t; rw [hXdef, hIdef, hZdef]; ring\n · intro t; rw [hZdef]; linarith [hcap t]\n · intro s t hst; rw [hIdef, hIdef]; linarith [hrate s t hst]\n · rw [hIdef]; linarith [hB0]\n · intro s t hst hb; rw [hIdef, hIdef]; linarith [hbusy s t hst hb]", "time_seconds": 40, "topic": "Continuous-time Markov & queues" }, { "question_id": 202, "problem_name": "stochastic_processes_strong_stationary_time_2", "informal_statement": "Let (X_n)_{n ≥ 0} be an irreducible Markov chain with stationary measure π.\n Let τ be a strong stationary time for (X_n).\n Let the separation distance be S_x (n) = max_y (1 − P^n (x, y) / π(y)).\n \n Show that S_x (n) ≤ ℙ_x [τ > n].", "lean_statement": "theorem q202_sst_sep {Ω' : Type*} [MeasurableSpace Ω'] (μ : Measure Ω') [IsProbabilityMeasure μ]\n {Ω : Type*} [Fintype Ω] [DecidableEq Ω] (P : Matrix Ω Ω ℝ) (π : Ω → ℝ) (hπpos : ∀ x, 0 < π x)\n (x0 : Ω) (X : ℕ → Ω' → Ω) (τ : Ω' → ℕ)\n (hmarg : HasMatrixMarginals μ X P x0)\n (hsst : IsStrongStationary μ X τ π)\n (n : ℕ) (y : Ω) :\n 1 - (P ^ n) x0 y / π y ≤ tailProb μ τ n := by sorry", "true_to_source": "abstract", "proof": "theorem q202_sst_sep {Ω' : Type*} [MeasurableSpace Ω'] (μ : Measure Ω') [IsProbabilityMeasure μ]\n {Ω : Type*} [Fintype Ω] [DecidableEq Ω] (P : Matrix Ω Ω ℝ) (π : Ω → ℝ) (hπpos : ∀ x, 0 < π x)\n (x0 : Ω) (X : ℕ → Ω' → Ω) (τ : Ω' → ℕ)\n (hmarg : HasMatrixMarginals μ X P x0)\n (hsst : IsStrongStationary μ X τ π)\n (n : ℕ) (y : Ω) :\n 1 - (P ^ n) x0 y / π y ≤ tailProb μ τ n := by\n have hp : 0 < π y := hπpos y\n have hfin1 : μ {ω | τ ω ≤ n} ≠ ⊤ := measure_ne_top μ _\n have hfin2 : μ {ω | n < τ ω} ≠ ⊤ := measure_ne_top μ _\n have hsub : {ω | τ ω ≤ n ∧ X n ω = y} ⊆ {ω | X n ω = y} := fun ω hω => hω.2\n have hmono : (μ {ω | τ ω ≤ n ∧ X n ω = y}).toReal ≤ (μ {ω | X n ω = y}).toReal :=\n ENNReal.toReal_mono (measure_ne_top μ _) (measure_mono hsub)\n rw [hsst n y, hmarg n y] at hmono\n have hle1 : (μ {ω | τ ω ≤ n}).toReal ≤ (P ^ n) x0 y / π y := by\n rw [le_div_iff₀ hp]; exact hmono\n have hcover : (Set.univ : Set Ω') ⊆ {ω | τ ω ≤ n} ∪ {ω | n < τ ω} := by\n intro ω _\n rcases Nat.lt_or_ge n (τ ω) with h | h\n · exact Or.inr h\n · exact Or.inl h\n have hsum : (1 : ℝ≥0∞) ≤ μ {ω | τ ω ≤ n} + μ {ω | n < τ ω} := by\n calc (1 : ℝ≥0∞) = μ (Set.univ : Set Ω') := by rw [measure_univ]\n _ ≤ μ ({ω | τ ω ≤ n} ∪ {ω | n < τ ω}) := measure_mono hcover\n _ ≤ μ {ω | τ ω ≤ n} + μ {ω | n < τ ω} := measure_union_le _ _\n have hsumR : (1 : ℝ) ≤ (μ {ω | τ ω ≤ n}).toReal + (μ {ω | n < τ ω}).toReal := by\n have h := ENNReal.toReal_mono (ENNReal.add_ne_top.mpr ⟨hfin1, hfin2⟩) hsum\n rwa [ENNReal.toReal_add hfin1 hfin2, ENNReal.toReal_one] at h\n show 1 - (P ^ n) x0 y / π y ≤ (μ {ω | n < τ ω}).toReal\n linarith", "time_seconds": 127, "topic": "Markov chains (finite & countable)" }, { "question_id": 206, "problem_name": "stochastic_processes_ctmc_guess_verify_reversal", "informal_statement": "Let (X(t)) be an irreducible continuous-time Markov chain with rates q_{ij}. Suppose a distribution (p_i) and candidate backward rates q*_{ij} satisfy Σ_j q_{ij} = Σ_j q*_{ij} and p_i q_{ij} = p_j q*_{ji} for all i, j.\n\nShow that (p_i) is the steady-state distribution and (q*_{ij}) are the transition rates of the time-reversed process.", "lean_statement": "theorem q206_ctmc_guess_verify_reversal {S : Type*} [Countable S]\n (q : S → S → ℝ) (hq0 : ∀ i j, 0 ≤ q i j) (hqloop : ∀ i, q i i = 0)\n (hqSum : ∀ i, Summable (q i))\n (qstar : S → S → ℝ) (hqs0 : ∀ i j, 0 ≤ qstar i j) (hqsloop : ∀ i, qstar i i = 0)\n (hqstarSum : ∀ i, Summable (qstar i))\n (p : S → ℝ) (hppos : ∀ i, 0 < p i) (hpsum : ∑' i, p i = 1)\n (hrowsum : ∀ i, ∑' j, q i j = ∑' j, qstar i j)\n (hbal : ∀ i j, p i * q i j = p j * qstar j i) :\n (∀ j, ∑' i, p i * q i j = p j * ∑' k, q j k) ∧\n (∀ i j, qstar i j = p j * q j i / p i) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q206_ctmc_guess_verify_reversal {S : Type*} [Countable S]\n (q : S → S → ℝ) (hq0 : ∀ i j, 0 ≤ q i j) (hqloop : ∀ i, q i i = 0)\n (hqSum : ∀ i, Summable (q i))\n (qstar : S → S → ℝ) (hqs0 : ∀ i j, 0 ≤ qstar i j) (hqsloop : ∀ i, qstar i i = 0)\n (hqstarSum : ∀ i, Summable (qstar i))\n (p : S → ℝ) (hppos : ∀ i, 0 < p i) (hpsum : ∑' i, p i = 1)\n (hrowsum : ∀ i, ∑' j, q i j = ∑' j, qstar i j)\n (hbal : ∀ i j, p i * q i j = p j * qstar j i) :\n (∀ j, ∑' i, p i * q i j = p j * ∑' k, q j k) ∧\n (∀ i j, qstar i j = p j * q j i / p i) := by\n refine ⟨?_, ?_⟩\n · intro j\n have h1 : ∀ i, p i * q i j = p j * qstar j i := fun i => hbal i j\n rw [tsum_congr h1, tsum_mul_left, ← hrowsum j]\n · intro i j\n have hpi : p i ≠ 0 := (hppos i).ne'\n rw [eq_div_iff hpi]\n linear_combination -(hbal j i)", "time_seconds": 65, "topic": "Continuous-time Markov & queues" }, { "question_id": 207, "problem_name": "stochastic_processes_mm1_process_geometric", "informal_statement": "Consider an M/M/1 queue as a continuous-time birth-death process with ρ = λ/μ < 1.\n\nShow that the fraction of time the system contains exactly i customers is (1 − ρ) ρ^i.", "lean_statement": "theorem q207_mm1_process_geometric\n (lam mu : ℝ) (hlam : 0 < lam) (hmu : 0 < mu)\n (ρ : ℝ) (hρ : ρ = lam / mu) (hρ0 : 0 < ρ) (hρ1 : ρ < 1)\n (p : ℕ → ℝ) (hp0 : ∀ i, 0 ≤ p i) (hpsum : ∑' i, p i = 1)\n (hbal0 : p 0 * lam = p 1 * mu)\n (hbal : ∀ n, p (n + 1) * (lam + mu) = p n * lam + p (n + 2) * mu) :\n ∀ i, p i = (1 - ρ) * ρ ^ i := by\n sorry", "true_to_source": "abstract", "proof": "theorem q207_mm1_process_geometric\n (lam mu : ℝ) (hlam : 0 < lam) (hmu : 0 < mu)\n (ρ : ℝ) (hρ : ρ = lam / mu) (hρ0 : 0 < ρ) (hρ1 : ρ < 1)\n (p : ℕ → ℝ) (hp0 : ∀ i, 0 ≤ p i) (hpsum : ∑' i, p i = 1)\n (hbal0 : p 0 * lam = p 1 * mu)\n (hbal : ∀ n, p (n + 1) * (lam + mu) = p n * lam + p (n + 2) * mu) :\n ∀ i, p i = (1 - ρ) * ρ ^ i := by\n have hmu0 : mu ≠ 0 := ne_of_gt hmu\n -- local (detailed) balance\n have hlb : ∀ n, p n * lam = p (n + 1) * mu := by\n intro n\n induction n with\n | zero => exact hbal0\n | succ m ih =>\n have h := hbal m\n have hexp : p (m + 1) * (lam + mu) = p (m + 1) * lam + p (m + 1) * mu := by ring\n linarith [h, ih, hexp]\n -- geometric step\n have hstep : ∀ n, p (n + 1) = p n * ρ := by\n intro n\n have h := (hlb n).symm\n rw [hρ, ← mul_div_assoc, eq_div_iff hmu0]\n exact h\n -- closed form up to p 0\n have hpow : ∀ n, p n = p 0 * ρ ^ n := by\n intro n\n induction n with\n | zero => simp\n | succ m ih => rw [hstep m, ih]; ring\n -- determine p 0\n have hgeo : ∑' i : ℕ, ρ ^ i = (1 - ρ)⁻¹ := tsum_geometric_of_lt_one (le_of_lt hρ0) hρ1\n have h1ρ : (1 - ρ) ≠ 0 := ne_of_gt (sub_pos.mpr hρ1)\n have hpsum' : ∑' i, p 0 * ρ ^ i = 1 := by\n rw [← hpsum]; exact tsum_congr (fun i => (hpow i).symm)\n rw [tsum_mul_left, hgeo] at hpsum'\n have hp0eq : p 0 = 1 - ρ := by\n field_simp at hpsum'\n linarith [hpsum']\n intro i\n rw [hpow i, hp0eq]", "time_seconds": 141, "topic": "Continuous-time Markov & queues" }, { "question_id": 210, "problem_name": "stochastic_processes_stopped_supermartingale_preserves", "informal_statement": "Let (Z_n) be a supermartingale and let J be a stopping trial with stopped process (Z_n*).\n\nShow that (Z_n*) is also a supermartingale.", "lean_statement": "theorem q210_stopped_supermartingale_preserves (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ]\n (Z : ℕ → Ω → ℝ) (hZ : Supermartingale Z ℱ μ)\n (J : Ω → ℕ) (hJ : IsStoppingTime ℱ (natStop J)) :\n Supermartingale (stoppedProcess Z (natStop J)) ℱ μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q210_stopped_supermartingale_preserves (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ]\n (Z : ℕ → Ω → ℝ) (hZ : Supermartingale Z ℱ μ)\n (J : Ω → ℕ) (hJ : IsStoppingTime ℱ (natStop J)) :\n Supermartingale (stoppedProcess Z (natStop J)) ℱ μ := by\n have h1 : Submartingale (stoppedProcess (-Z) (natStop J)) ℱ μ := hZ.neg.stoppedProcess hJ\n have h3 : -(stoppedProcess (-Z) (natStop J)) = stoppedProcess Z (natStop J) := by\n funext i ω\n simp [MeasureTheory.stoppedProcess]\n have h2 := h1.neg\n rwa [h3] at h2", "time_seconds": 777, "topic": "Martingales & stopping" }, { "question_id": 212, "problem_name": "stochastic_processes_jackson_reverse_process", "informal_statement": "Consider an open Jackson network in steady state.\n\nShow that the time-reversed process is itself a Jackson network, with service rates unchanged and routing probabilities Q*_{ij} = λ_j Q_{ji} / λ_i.", "lean_statement": "theorem q212_jackson_reverse_process\n (k : ℕ) (γ : Fin k → ℝ) (Q : Fin k → Fin k → ℝ) (Q0 : Fin k → ℝ)\n (hQ : ∀ i j, 0 ≤ Q i j) (hcons : ∀ i, Q0 i + ∑ j, Q i j = 1)\n (lam : Fin k → ℝ) (hlam : ∀ i, 0 < lam i)\n (htraffic : ∀ j, lam j = γ j + ∑ i, lam i * Q i j)\n (Qstar : Fin k → Fin k → ℝ) (hQstar : ∀ i j, Qstar i j = lam j * Q j i / lam i) :\n (∀ i j, 0 ≤ Qstar i j) ∧\n (∀ i, γ i / lam i + ∑ j, Qstar i j = 1) ∧\n (∀ i, lam i = lam i * Q0 i + ∑ j, lam j * Qstar j i) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q212_jackson_reverse_process\n (k : ℕ) (γ : Fin k → ℝ) (Q : Fin k → Fin k → ℝ) (Q0 : Fin k → ℝ)\n (hQ : ∀ i j, 0 ≤ Q i j) (hcons : ∀ i, Q0 i + ∑ j, Q i j = 1)\n (lam : Fin k → ℝ) (hlam : ∀ i, 0 < lam i)\n (htraffic : ∀ j, lam j = γ j + ∑ i, lam i * Q i j)\n (Qstar : Fin k → Fin k → ℝ) (hQstar : ∀ i j, Qstar i j = lam j * Q j i / lam i) :\n (∀ i j, 0 ≤ Qstar i j) ∧\n (∀ i, γ i / lam i + ∑ j, Qstar i j = 1) ∧\n (∀ i, lam i = lam i * Q0 i + ∑ j, lam j * Qstar j i) := by\n refine ⟨?_, ?_, ?_⟩\n · -- nonnegativity\n intro i j\n rw [hQstar i j]\n exact div_nonneg (mul_nonneg (le_of_lt (hlam j)) (hQ j i)) (le_of_lt (hlam i))\n · -- row sums of Qstar\n intro i\n have hli : lam i ≠ 0 := ne_of_gt (hlam i)\n have ht := htraffic i\n have hsum : ∑ j, Qstar i j = (∑ j, lam j * Q j i) / lam i := by\n rw [Finset.sum_div]\n exact Finset.sum_congr rfl (fun j _ => hQstar i j)\n rw [hsum]\n field_simp\n linarith [ht]\n · -- reversed traffic equations\n intro i\n have hc := hcons i\n have key : ∀ j, lam j * Qstar j i = lam i * Q i j := by\n intro j\n rw [hQstar j i]\n field_simp [(hlam j).ne']\n have hsum3 : ∑ j, lam j * Qstar j i = lam i * ∑ j, Q i j := by\n rw [Finset.mul_sum]\n exact Finset.sum_congr rfl (fun j _ => key j)\n rw [hsum3]\n have hqi : ∑ j, Q i j = 1 - Q0 i := by linarith [hc]\n rw [hqi]; ring", "time_seconds": 167, "topic": "Continuous-time Markov & queues" }, { "question_id": 213, "problem_name": "stochastic_processes_aperiodicity", "informal_statement": "Suppose that P is irreducible and aperiodic.\n \n Show that there exists an integer r such that P^n (x, y) > 0, ∀x,y ∈ Ω, ∀n ≥ r.", "lean_statement": "theorem q213_aperiodicity {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (hirr : IsIrreducible P)\n (haper : IsAperiodic P) :\n IsPrimitive P := by sorry", "true_to_source": "literal", "proof": "theorem q213_aperiodicity {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (hirr : IsIrreducible P)\n (haper : IsAperiodic P) :\n IsPrimitive P := by\n have hPnn : ∀ n : ℕ, ∀ i j : Ω, 0 ≤ (P ^ n) i j := by\n intro n\n induction n with\n | zero =>\n intro i j\n rw [pow_zero, Matrix.one_apply]\n split <;> norm_num\n | succ k ih =>\n intro i j\n rw [pow_succ, Matrix.mul_apply]\n exact Finset.sum_nonneg (fun z _ => mul_nonneg (ih i z) (hP.1 z j))\n have key : ∀ (a b : ℕ) (i j : Ω), (P ^ a) i i * (P ^ b) i j ≤ (P ^ (a + b)) i j := by\n intro a b i j\n rw [pow_add, Matrix.mul_apply]\n exact Finset.single_le_sum (f := fun z => (P ^ a) i z * (P ^ b) z j)\n (fun z _ => mul_nonneg (hPnn a i z) (hPnn b z j)) (Finset.mem_univ i)\n have hdiag : ∀ x : Ω, ∃ N : ℕ, ∀ n : ℕ, N ≤ n → 0 < (P ^ n) x x := by\n intro x\n set S : Set ℕ := {n | 0 < (P ^ n) x x} with hS\n have hzero : (0 : ℕ) ∈ S := by\n rw [hS]; change 0 < (P ^ 0) x x\n rw [pow_zero, Matrix.one_apply_eq]; norm_num\n have hadd : ∀ a ∈ S, ∀ b ∈ S, a + b ∈ S := by\n intro a ha b hb\n have hle : (P ^ a) x x * (P ^ b) x x ≤ (P ^ (a + b)) x x := key a b x x\n have ha' : 0 < (P ^ a) x x := ha\n have hb' : 0 < (P ^ b) x x := hb\n exact lt_of_lt_of_le (mul_pos ha' hb') hle\n let M : AddSubmonoid ℕ :=\n { carrier := S\n zero_mem' := hzero\n add_mem' := fun {a b} ha hb => hadd a ha b hb }\n have hclosure : AddSubmonoid.closure S ≤ M := AddSubmonoid.closure_le.mpr subset_rfl\n have hdvd_mem : ∀ n : ℕ, 0 < (P ^ n) x x → Nat.setGcd S ∣ n := by\n intro n hn\n have hnS : n ∈ S := by rw [hS]; exact hn\n exact Nat.setGcd_dvd_of_mem_closure (AddSubmonoid.subset_closure hnS)\n have hgcd_le : Nat.setGcd S ≤ 1 := haper x (Nat.setGcd S) (fun n hn => hdvd_mem n hn)\n have hgcd_ne : Nat.setGcd S ≠ 0 := by\n obtain ⟨n₀, hn₀, hpos⟩ := hirr x x\n have hd : Nat.setGcd S ∣ n₀ := hdvd_mem n₀ hpos\n intro h0\n rw [h0] at hd\n have : n₀ = 0 := Nat.eq_zero_of_zero_dvd hd\n omega\n have hgcd : Nat.setGcd S = 1 := by\n have : 1 ≤ Nat.setGcd S := Nat.one_le_iff_ne_zero.mpr hgcd_ne\n omega\n obtain ⟨N, hN⟩ := Nat.exists_mem_closure_of_ge S\n refine ⟨N, fun n hn => ?_⟩\n have hmem : n ∈ AddSubmonoid.closure S := hN n hn (by rw [hgcd]; exact one_dvd n)\n have hnM : n ∈ S := hclosure hmem\n rw [hS] at hnM\n exact hnM\n choose Nfun hNfun using hdiag\n have hirr' : ∀ x y : Ω, ∃ m : ℕ, 1 ≤ m ∧ 0 < (P ^ m) x y := hirr\n choose mfun hmfun1 hmfun2 using hirr'\n refine ⟨Finset.univ.sup Nfun + Finset.univ.sup (fun p : Ω × Ω => mfun p.1 p.2),\n fun x y n hn => ?_⟩\n have hm_le : mfun x y ≤ Finset.univ.sup (fun p : Ω × Ω => mfun p.1 p.2) := by\n have := Finset.le_sup (f := fun p : Ω × Ω => mfun p.1 p.2) (Finset.mem_univ (x, y))\n simpa using this\n have hNx_le : Nfun x ≤ Finset.univ.sup Nfun := Finset.le_sup (Finset.mem_univ x)\n have hdiff : Nfun x ≤ n - mfun x y := by omega\n have hpos_diag : 0 < (P ^ (n - mfun x y)) x x := hNfun x (n - mfun x y) hdiff\n have hpos_off : 0 < (P ^ (mfun x y)) x y := hmfun2 x y\n have hsplit : (n - mfun x y) + mfun x y = n := by omega\n have hk := key (n - mfun x y) (mfun x y) x y\n rw [hsplit] at hk\n exact lt_of_lt_of_le (mul_pos hpos_diag hpos_off) hk", "time_seconds": 197, "topic": "Markov chains (finite & countable)" }, { "question_id": 214, "problem_name": "stochastic_processes_irreducibility", "informal_statement": "Let P be irreducible.\n \n Show that gcd(T(x)) = gcd(T(y)) for all x, y ∈ Ω.", "lean_statement": "theorem q214_irreducibility_gcd {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (hirr : IsIrreducible P) :\n ∀ x y : Ω, ∀ d : ℕ,\n (∀ n, 0 < (P ^ n) x x → d ∣ n) ↔ (∀ n, 0 < (P ^ n) y y → d ∣ n) := by sorry", "true_to_source": "literal", "proof": "theorem q214_irreducibility_gcd {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (hirr : IsIrreducible P) :\n ∀ x y : Ω, ∀ d : ℕ,\n (∀ n, 0 < (P ^ n) x x → d ∣ n) ↔ (∀ n, 0 < (P ^ n) y y → d ∣ n) := by\n have hnn : ∀ n : ℕ, ∀ a b, 0 ≤ (P ^ n) a b := by\n intro n\n induction n with\n | zero => intro a b; rw [pow_zero, Matrix.one_apply]; split <;> norm_num\n | succ k ih =>\n intro a b\n rw [pow_succ, Matrix.mul_apply]\n exact Finset.sum_nonneg (fun z _ => mul_nonneg (ih a z) (hP.1 z b))\n have hmul : ∀ (m n : ℕ) (a w b : Ω),\n 0 < (P ^ m) a w → 0 < (P ^ n) w b → 0 < (P ^ (m + n)) a b := by\n intro m n a w b haw hwb\n rw [pow_add, Matrix.mul_apply]\n refine lt_of_lt_of_le (mul_pos haw hwb) ?_\n exact Finset.single_le_sum (f := fun z => (P ^ m) a z * (P ^ n) z b)\n (fun z _ => mul_nonneg (hnn m a z) (hnn n z b)) (Finset.mem_univ w)\n intro x y d\n have helper : ∀ u v : Ω,\n (∀ n, 0 < (P ^ n) u u → d ∣ n) → ∀ m, 0 < (P ^ m) v v → d ∣ m := by\n intro u v Hu m hm\n obtain ⟨a, _, ha⟩ := hirr v u\n obtain ⟨b, _, hb⟩ := hirr u v\n have hbaU : 0 < (P ^ (b + a)) u u := hmul b a u v u hb ha\n have hbmU : 0 < (P ^ (b + m)) u v := hmul b m u v v hb hm\n have hbmaU : 0 < (P ^ ((b + m) + a)) u u := hmul (b + m) a u v u hbmU ha\n have d1 : d ∣ (b + a) := Hu (b + a) hbaU\n have d2 : d ∣ ((b + a) + m) := by\n have heq : (b + m) + a = (b + a) + m := by ring\n rw [heq] at hbmaU\n exact Hu ((b + a) + m) hbmaU\n exact (Nat.dvd_add_right d1).mp d2\n exact ⟨helper x y, helper y x⟩", "time_seconds": 54, "topic": "Markov chains (finite & countable)" }, { "question_id": 217, "problem_name": "stochastic_processes_doob_decomposition", "informal_statement": "Let (X_n) be a submartingale adapted to a filtration (F_n).\n\nShow that X_n admits a decomposition X_n = M_n + A_n where (M_n) is a martingale and (A_n) is almost surely non-decreasing and predictable, and that this decomposition is unique.", "lean_statement": "theorem q217_doob_decomposition (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (X : ℕ → Ω → ℝ) (hX : Submartingale X ℱ μ) :\n ∃ M A : ℕ → Ω → ℝ,\n Martingale M ℱ μ ∧\n IsStronglyPredictable ℱ A ∧\n A 0 = 0 ∧\n (∀ᵐ ω ∂μ, Monotone fun n => A n ω) ∧\n (∀ n, X n =ᵐ[μ] fun ω => M n ω + A n ω) ∧\n (∀ M' A' : ℕ → Ω → ℝ, Martingale M' ℱ μ → IsStronglyPredictable ℱ A' → A' 0 = 0 →\n (∀ n, X n =ᵐ[μ] fun ω => M' n ω + A' n ω) →\n (∀ n, M n =ᵐ[μ] M' n) ∧ (∀ n, A n =ᵐ[μ] A' n)) := by\n sorry", "true_to_source": "literal", "proof": "theorem q217_doob_decomposition (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (X : ℕ → Ω → ℝ) (hX : Submartingale X ℱ μ) :\n ∃ M A : ℕ → Ω → ℝ,\n Martingale M ℱ μ ∧\n IsStronglyPredictable ℱ A ∧\n A 0 = 0 ∧\n (∀ᵐ ω ∂μ, Monotone fun n => A n ω) ∧\n (∀ n, X n =ᵐ[μ] fun ω => M n ω + A n ω) ∧\n (∀ M' A' : ℕ → Ω → ℝ, Martingale M' ℱ μ → IsStronglyPredictable ℱ A' → A' 0 = 0 →\n (∀ n, X n =ᵐ[μ] fun ω => M' n ω + A' n ω) →\n (∀ n, M n =ᵐ[μ] M' n) ∧ (∀ n, A n =ᵐ[μ] A' n)) := by\n refine ⟨martingalePart X ℱ μ, predictablePart X ℱ μ, ?_, ?_, ?_, ?_, ?_, ?_⟩\n · exact martingale_martingalePart hX.stronglyAdapted hX.integrable\n · exact isPredictable_predictablePart\n · exact predictablePart_zero\n · exact hX.monotone_predictablePart\n · intro n\n have h := congrFun (martingalePart_add_predictablePart ℱ μ X) n\n filter_upwards with ω\n have := congrFun h ω\n simpa [Pi.add_apply] using this.symm\n · intro M' A' hM' hA' hA'0 hXeq'\n -- Integrability of A'\n have hA'int : ∀ n, Integrable (A' n) μ := by\n intro n\n have heq : (fun ω => X n ω - M' n ω) =ᵐ[μ] A' n := by\n filter_upwards [hXeq' n] with ω h\n rw [h]; ring\n exact ((hX.integrable n).sub (hM'.integrable n)).congr heq\n -- predictablePart X = predictablePart (M' + A')\n have hpp : ∀ n, predictablePart X ℱ μ n =ᵐ[μ] predictablePart (M' + A') ℱ μ n := by\n intro n\n simp only [predictablePart]\n refine eventuallyEq_sum (fun i _ => ?_)\n apply condExp_congr_ae\n filter_upwards [hXeq' (i + 1), hXeq' i] with ω h1 h2\n simp only [Pi.sub_apply, Pi.add_apply]\n rw [h1, h2]\n -- martingalePart X = martingalePart (M' + A')\n have hmp : ∀ n, martingalePart X ℱ μ n =ᵐ[μ] martingalePart (M' + A') ℱ μ n := by\n intro n\n simp only [martingalePart]\n filter_upwards [hXeq' n, hpp n] with ω hx hp\n simp only [Pi.sub_apply, Pi.add_apply]\n rw [hx, hp]\n refine ⟨fun n => ?_, fun n => ?_⟩\n · exact (hmp n).trans (martingalePart_add_ae_eq hM' hA'.measurable_add_one hA'0 hA'int n)\n · exact (hpp n).trans (predictablePart_add_ae_eq hM' hA'.measurable_add_one hA'0 hA'int n)", "time_seconds": 235, "topic": "Martingales & stopping" }, { "question_id": 218, "problem_name": "stochastic_processes_strong_stationary_time_4", "informal_statement": "Let (X_n)_{n ≥ 0} be an irreducible Markov chain with stationary measure π.\n Let τ be a strong stationary time for (X_n).\n \n Show that max_x ||P^n (x, ·) − π||_{TV} ≤ max_x ℙ_x [τ > n].", "lean_statement": "theorem q218_sst_tv {Ω' : Type*} [MeasurableSpace Ω'] (μ : Measure Ω') [IsProbabilityMeasure μ]\n {Ω : Type*} [Fintype Ω] [DecidableEq Ω] (P : Matrix Ω Ω ℝ) (hP : IsStochastic P)\n (π : Ω → ℝ) (hπ : IsStationary P π) (x0 : Ω) (X : ℕ → Ω' → Ω) (τ : Ω' → ℕ)\n (hmarg : HasMatrixMarginals μ X P x0)\n (hsst : IsStrongStationary μ X τ π)\n (n : ℕ) :\n tvDist (fun y => (P ^ n) x0 y) π ≤ tailProb μ τ n := by sorry", "true_to_source": "abstract", "proof": "theorem q218_sst_tv {Ω' : Type*} [MeasurableSpace Ω'] (μ : Measure Ω') [IsProbabilityMeasure μ]\n {Ω : Type*} [Fintype Ω] [DecidableEq Ω] (P : Matrix Ω Ω ℝ) (hP : IsStochastic P)\n (π : Ω → ℝ) (hπ : IsStationary P π) (x0 : Ω) (X : ℕ → Ω' → Ω) (τ : Ω' → ℕ)\n (hmarg : HasMatrixMarginals μ X P x0)\n (hsst : IsStrongStationary μ X τ π)\n (n : ℕ) :\n tvDist (fun y => (P ^ n) x0 y) π ≤ tailProb μ τ n := by\n -- row sums of stochastic power\n have hrow : ∀ (m : ℕ) (a : Ω), ∑ b, (P ^ m) a b = 1 := by\n intro m\n induction m with\n | zero => intro a; simp [pow_zero, Matrix.one_apply, Finset.sum_ite_eq]\n | succ k ih =>\n intro a\n rw [pow_succ]\n simp_rw [Matrix.mul_apply]\n rw [Finset.sum_comm]\n simp_rw [← Finset.mul_sum, hP.2, mul_one]\n exact ih a\n simp only [Auto.tailProb]\n set a : Ω → ℝ := fun y => (P ^ n) x0 y with ha_def\n set s : ℝ := (μ {ω | τ ω ≤ n}).toReal with hs_def\n set t : ℝ := (μ {ω | n < τ ω}).toReal with ht_def\n have hasum : ∑ y, a y = 1 := hrow n x0\n have hbsum : ∑ y, π y = 1 := hπ.2.1\n have hbnn : ∀ y, 0 ≤ π y := hπ.1\n have ht_nonneg : 0 ≤ t := ENNReal.toReal_nonneg\n -- 1 ≤ s + t\n have hst : 1 ≤ s + t := by\n have hcover : (Set.univ : Set Ω') ⊆ {ω | τ ω ≤ n} ∪ {ω | n < τ ω} := by\n intro ω _\n by_cases h : τ ω ≤ n\n · exact Or.inl h\n · exact Or.inr (not_le.mp h)\n have hle : μ (Set.univ) ≤ μ {ω | τ ω ≤ n} + μ {ω | n < τ ω} :=\n le_trans (measure_mono hcover) (measure_union_le _ _)\n have hfin1 : μ {ω | τ ω ≤ n} ≠ ⊤ := measure_ne_top μ _\n have hfin2 : μ {ω | n < τ ω} ≠ ⊤ := measure_ne_top μ _\n have hmono := ENNReal.toReal_mono (ENNReal.add_ne_top.mpr ⟨hfin1, hfin2⟩) hle\n rw [ENNReal.toReal_add hfin1 hfin2] at hmono\n simpa [measure_univ] using hmono\n -- lower bound a y ≥ s * π y\n have hlb : ∀ y, s * π y ≤ a y := by\n intro y\n have hsub : {ω | τ ω ≤ n ∧ X n ω = y} ⊆ {ω | X n ω = y} := fun ω h => h.2\n have hmono : (μ {ω | τ ω ≤ n ∧ X n ω = y}).toReal ≤ (μ {ω | X n ω = y}).toReal :=\n ENNReal.toReal_mono (measure_ne_top μ _) (measure_mono hsub)\n rw [hsst n y, hmarg n y] at hmono\n exact hmono\n -- TV identity\n have hidentity : ∑ y, |a y - π y| = 2 * ∑ y, max (π y - a y) 0 := by\n have e1 : ∀ y, |a y - π y| = 2 * max (π y - a y) 0 - (π y - a y) := by\n intro y\n rw [abs_sub_comm]\n rcases le_total 0 (π y - a y) with h | h\n · rw [abs_of_nonneg h, max_eq_left h]; ring\n · rw [abs_of_nonpos h, max_eq_right h]; ring\n rw [Finset.sum_congr rfl (fun y _ => e1 y), Finset.sum_sub_distrib, ← Finset.mul_sum]\n have hz : ∑ y, (π y - a y) = 0 := by\n rw [Finset.sum_sub_distrib, hbsum, hasum]; ring\n rw [hz]; ring\n -- per-term bound\n have hmax_le : ∀ y, max (π y - a y) 0 ≤ t * π y := by\n intro y\n rcases le_total (π y - a y) 0 with h | h\n · rw [max_eq_right h]; exact mul_nonneg ht_nonneg (hbnn y)\n · rw [max_eq_left h]\n have h1 : π y - a y ≤ (1 - s) * π y := by\n have he : (1 - s) * π y = π y - s * π y := by ring\n rw [he]; linarith [hlb y]\n have h2 : (1 - s) * π y ≤ t * π y :=\n mul_le_mul_of_nonneg_right (by linarith [hst]) (hbnn y)\n linarith\n -- assemble\n simp only [Auto.tvDist]\n calc (1/2) * ∑ y, |a y - π y|\n = (1/2) * (2 * ∑ y, max (π y - a y) 0) := by rw [hidentity]\n _ = ∑ y, max (π y - a y) 0 := by ring\n _ ≤ ∑ y, t * π y := Finset.sum_le_sum (fun y _ => hmax_le y)\n _ = t * (∑ y, π y) := by rw [← Finset.mul_sum]\n _ = t := by rw [hbsum]; ring", "time_seconds": 61, "topic": "Markov chains (finite & countable)" }, { "question_id": 220, "problem_name": "stochastic_processes_poisson_conditional_two_intervals", "informal_statement": "Let (N(t)) be a Poisson process of rate λ and let 0 < s < t.\n\nShow that the conditional distribution of N(s) given N(t) = n is binomial with parameters n and s/t.", "lean_statement": "theorem q220_poisson_conditional_binomial (μ : Measure Ω) [IsProbabilityMeasure μ]\n (lam s t : ℝ≥0) (hlam : 0 < lam) (hs : 0 < s) (hst : s < t)\n (N : ℝ≥0 → Ω → ℕ)\n (hNs : HasLaw (N s) (poissonMeasure (lam * s)) μ)\n (hNincr : HasLaw (fun ω => N t ω - N s ω) (poissonMeasure (lam * (t - s))) μ)\n (hindep : IndepFun (N s) (fun ω => N t ω - N s ω) μ)\n (hmono : ∀ ω, N s ω ≤ N t ω) :\n ∀ n k : ℕ, k ≤ n →\n (μ {ω | N s ω = k ∧ N t ω = n}).toReal\n = (μ {ω | N t ω = n}).toReal\n * (Nat.choose n k : ℝ) * ((s : ℝ) / t) ^ k * (1 - (s : ℝ) / t) ^ (n - k) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q220_poisson_conditional_binomial (μ : Measure Ω) [IsProbabilityMeasure μ]\n (lam s t : ℝ≥0) (hlam : 0 < lam) (hs : 0 < s) (hst : s < t)\n (N : ℝ≥0 → Ω → ℕ)\n (hNs : HasLaw (N s) (poissonMeasure (lam * s)) μ)\n (hNincr : HasLaw (fun ω => N t ω - N s ω) (poissonMeasure (lam * (t - s))) μ)\n (hindep : IndepFun (N s) (fun ω => N t ω - N s ω) μ)\n (hmono : ∀ ω, N s ω ≤ N t ω) :\n ∀ n k : ℕ, k ≤ n →\n (μ {ω | N s ω = k ∧ N t ω = n}).toReal\n = (μ {ω | N t ω = n}).toReal\n * (Nat.choose n k : ℝ) * ((s : ℝ) / t) ^ k * (1 - (s : ℝ) / t) ^ (n - k) := by\n intro n k hkn\n -- Law of N t\n have hrate : lam * s + lam * (t - s) = lam * t := by\n rw [← mul_add, add_tsub_cancel_of_le hst.le]\n have hadd := hindep.hasLaw_add_poissonMeasure hNs hNincr\n rw [hrate] at hadd\n have hNt_law : HasLaw (N t) (poissonMeasure (lam * t)) μ := by\n refine hadd.congr ?_\n filter_upwards with ω\n simp only [Pi.add_apply]\n exact (Nat.add_sub_cancel' (hmono ω)).symm\n -- Event decomposition\n have hAeq : {ω | N s ω = k ∧ N t ω = n}\n = (N s) ⁻¹' {k} ∩ (fun ω => N t ω - N s ω) ⁻¹' {n - k} := by\n ext ω\n simp only [Set.mem_setOf_eq, Set.mem_inter_iff, Set.mem_preimage, Set.mem_singleton_iff]\n constructor\n · rintro ⟨hk, hn⟩\n exact ⟨hk, by rw [hk, hn]⟩\n · rintro ⟨hk, hd⟩\n refine ⟨hk, ?_⟩\n have hh := Nat.add_sub_cancel' (hmono ω)\n rw [hd, hk, Nat.add_sub_cancel' hkn] at hh\n exact hh.symm\n -- Independence product\n have hprodmeas := hindep.measure_inter_preimage_eq_mul {k} {n - k}\n (measurableSet_singleton k) (measurableSet_singleton (n - k))\n -- pmf values (clean form)\n have hsk : (μ ((N s) ⁻¹' {k})).toReal = (poissonMeasure (lam * s)).real {k} := by\n have hpre : ((N s) ⁻¹' {k}) = {ω | N s ω = k} := by\n ext ω; simp\n rw [hpre, ← measureReal_def,\n hNs.measureReal_eq (p := fun x => x = k)\n (by rw [Set.setOf_eq_eq_singleton]; exact measurableSet_singleton k),\n Set.setOf_eq_eq_singleton]\n have hincr : (μ ((fun ω => N t ω - N s ω) ⁻¹' {n - k})).toReal\n = (poissonMeasure (lam * (t - s))).real {n - k} := by\n have hpre : ((fun ω => N t ω - N s ω) ⁻¹' {n - k}) = {ω | N t ω - N s ω = n - k} := by\n ext ω; simp\n rw [hpre, ← measureReal_def,\n hNincr.measureReal_eq (p := fun x => x = n - k)\n (by rw [Set.setOf_eq_eq_singleton]; exact measurableSet_singleton (n - k)),\n Set.setOf_eq_eq_singleton]\n have htn : (μ {ω | N t ω = n}).toReal = (poissonMeasure (lam * t)).real {n} := by\n rw [← measureReal_def,\n hNt_law.measureReal_eq (p := fun x => x = n)\n (by rw [Set.setOf_eq_eq_singleton]; exact measurableSet_singleton n),\n Set.setOf_eq_eq_singleton]\n -- combine\n rw [hAeq, show (μ ((N s) ⁻¹' {k} ∩ (fun ω => N t ω - N s ω) ⁻¹' {n - k})).toReal\n = (μ ((N s) ⁻¹' {k})).toReal * (μ ((fun ω => N t ω - N s ω) ⁻¹' {n - k})).toReal from by\n rw [hprodmeas, ENNReal.toReal_mul], hsk, hincr, htn,\n poissonMeasure_real_singleton, poissonMeasure_real_singleton, poissonMeasure_real_singleton]\n -- real algebraic identity\n have hSpos : (0:ℝ) < s := by exact_mod_cast hs\n have hST : (s:ℝ) < t := by exact_mod_cast hst\n have hTpos : (0:ℝ) < t := lt_trans hSpos hST\n have hchoose : (Nat.choose n k : ℝ) * (Nat.factorial k : ℝ) * (Nat.factorial (n - k) : ℝ)\n = (Nat.factorial n : ℝ) := by\n exact_mod_cast Nat.choose_mul_factorial_mul_factorial hkn\n have hcpos : (Nat.choose n k : ℝ) ≠ 0 := by\n exact_mod_cast (Nat.choose_pos hkn).ne'\n have hexp : Real.exp (-((lam:ℝ) * s)) * Real.exp (-((lam:ℝ) * ((t:ℝ) - s)))\n = Real.exp (-((lam:ℝ) * t)) := by\n rw [← Real.exp_add]; ring_nf\n have hLpow : (lam:ℝ) ^ n = (lam:ℝ) ^ k * (lam:ℝ) ^ (n - k) := by\n rw [← pow_add, Nat.add_sub_cancel' hkn]\n have hTpow : (t:ℝ) ^ n = (t:ℝ) ^ k * (t:ℝ) ^ (n - k) := by\n rw [← pow_add, Nat.add_sub_cancel' hkn]\n push_cast [NNReal.coe_sub hst.le]\n simp only [mul_pow]\n rw [← hexp, hLpow, hTpow,\n show (1 - (s:ℝ)/t) = ((t:ℝ) - s)/t from by field_simp,\n div_pow, div_pow, ← hchoose]\n field_simp", "time_seconds": 368, "topic": "Poisson processes" }, { "question_id": 221, "problem_name": "stochastic_processes_finite_chain_aperiodic_self_loop", "informal_statement": "Let (X_n) be an irreducible finite-state Markov chain in which some state i has P_{ii} > 0.\n\nShow that the chain is aperiodic.", "lean_statement": "theorem q221_finite_chain_aperiodic_self_loop {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (hirr : IsIrreducible P)\n (i : Ω) (hself : 0 < P i i) :\n IsAperiodic P := by\n sorry", "true_to_source": "abstract", "proof": "theorem q221_finite_chain_aperiodic_self_loop {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (hirr : IsIrreducible P)\n (i : Ω) (hself : 0 < P i i) :\n IsAperiodic P := by\n have hnn : ∀ (n : ℕ) (u v : Ω), 0 ≤ (P ^ n) u v := by\n intro n\n induction n with\n | zero =>\n intro u v\n rw [pow_zero, Matrix.one_apply]\n split <;> norm_num\n | succ k ih =>\n intro u v\n rw [pow_succ, Matrix.mul_apply]\n exact Finset.sum_nonneg (fun z _ => mul_nonneg (ih u z) (hP.1 z v))\n have hstep : ∀ (s t : ℕ) (u w z : Ω), (P ^ s) u z * (P ^ t) z w ≤ (P ^ (s + t)) u w := by\n intro s t u w z\n rw [pow_add, Matrix.mul_apply]\n exact Finset.single_le_sum (f := fun z' => (P ^ s) u z' * (P ^ t) z' w)\n (fun z' _ => mul_nonneg (hnn s u z') (hnn t z' w)) (Finset.mem_univ z)\n intro x d hd\n obtain ⟨a, _, ha⟩ := hirr x i\n obtain ⟨b, _, hb⟩ := hirr i x\n have hself1 : 0 < (P ^ 1) i i := by rw [pow_one]; exact hself\n have hab : 0 < (P ^ (a + b)) x x :=\n lt_of_lt_of_le (mul_pos ha hb) (hstep a b x x i)\n have hb1 : 0 < (P ^ (1 + b)) i x :=\n lt_of_lt_of_le (mul_pos hself1 hb) (hstep 1 b i x i)\n have hab1 : 0 < (P ^ (a + (1 + b))) x x :=\n lt_of_lt_of_le (mul_pos ha hb1) (hstep a (1 + b) x x i)\n have h1 := hd (a + b) hab\n have h2 := hd (a + (1 + b)) hab1\n have hd1 : d ∣ 1 := by\n have hsub := Nat.dvd_sub h2 h1\n have he : a + (1 + b) - (a + b) = 1 := by omega\n rwa [he] at hsub\n exact Nat.le_of_dvd one_pos hd1", "time_seconds": 197, "topic": "Markov chains (finite & countable)" }, { "question_id": 222, "problem_name": "stochastic_processes_stationary_existance_1", "informal_statement": "Suppose that P is irreducible.\n\n Show that there exists a probability measure π such that π = πP and π(x) > 0 for all x ∈ Ω.", "lean_statement": "theorem q222_stationary_existence_positive {Ω : Type*} [Fintype Ω] [DecidableEq Ω] [Nonempty Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (hirr : IsIrreducible P) :\n ∃ π : Ω → ℝ, IsStationary P π ∧ ∀ x, 0 < π x := by sorry", "true_to_source": "literal", "proof": "private def stepMap {Ω : Type*} [Fintype Ω] (P : Matrix Ω Ω ℝ) (v : Ω → ℝ) : Ω → ℝ :=\n fun y => ∑ x, v x * P x y\n\n/-- Cesàro averages of the iterates of `stepMap P` started at `v0`. -/\nprivate noncomputable def cesaro {Ω : Type*} [Fintype Ω] (P : Matrix Ω Ω ℝ) (v0 : Ω → ℝ)\n (N : ℕ) : Ω → ℝ :=\n fun y => (1 / (N + 1 : ℝ)) * ∑ n ∈ Finset.range (N + 1), ((stepMap P)^[n] v0) y\n\n/-- `stepMap P` maps the standard simplex into itself. -/\nprivate lemma stepMap_mem {Ω : Type*} [Fintype Ω] {P : Matrix Ω Ω ℝ}\n (hP0 : ∀ x y, 0 ≤ P x y) (hP1 : ∀ x, ∑ y, P x y = 1)\n {v : Ω → ℝ} (hv : v ∈ stdSimplex ℝ Ω) : stepMap P v ∈ stdSimplex ℝ Ω := by\n refine ⟨fun y => ?_, ?_⟩\n · show 0 ≤ ∑ x, v x * P x y\n exact Finset.sum_nonneg (fun x _ => mul_nonneg (hv.1 x) (hP0 x y))\n · show ∑ y, ∑ x, v x * P x y = 1\n rw [Finset.sum_comm]\n calc ∑ x, ∑ y, v x * P x y = ∑ x, v x * ∑ y, P x y := by\n apply Finset.sum_congr rfl; intro x _; rw [Finset.mul_sum]\n _ = ∑ x, v x * 1 := by apply Finset.sum_congr rfl; intro x _; rw [hP1 x]\n _ = ∑ x, v x := by apply Finset.sum_congr rfl; intro x _; rw [mul_one]\n _ = 1 := hv.2\n\n/-- Entries of powers of a nonnegative matrix are nonnegative. -/\nprivate lemma pow_nonneg_entries {Ω : Type*} [Fintype Ω] [DecidableEq Ω] {P : Matrix Ω Ω ℝ}\n (hP0 : ∀ x y, 0 ≤ P x y) : ∀ n x y, 0 ≤ (P ^ n) x y := by\n intro n\n induction n with\n | zero => intro x y; rw [pow_zero]; by_cases h : x = y <;> simp [Matrix.one_apply, h]\n | succ n ih =>\n intro x y\n rw [pow_succ, Matrix.mul_apply]\n exact Finset.sum_nonneg (fun z _ => mul_nonneg (ih x z) (hP0 z y))\n\n/-- A stationary distribution is stationary for every power of `P`. -/\nprivate lemma stationary_pow {Ω : Type*} [Fintype Ω] [DecidableEq Ω] {P : Matrix Ω Ω ℝ}\n {π : Ω → ℝ} (hstat : ∀ y, ∑ x, π x * P x y = π y) :\n ∀ n y, ∑ x, π x * (P ^ n) x y = π y := by\n intro n\n induction n with\n | zero =>\n intro y\n rw [pow_zero]\n simp [Matrix.one_apply, Finset.sum_ite_eq']\n | succ n ih =>\n intro y\n calc ∑ x, π x * (P ^ (n + 1)) x y\n = ∑ x, ∑ z, π x * ((P ^ n) x z * P z y) := by\n apply Finset.sum_congr rfl; intro x _\n rw [pow_succ, Matrix.mul_apply, Finset.mul_sum]\n _ = ∑ z, ∑ x, π x * ((P ^ n) x z * P z y) := Finset.sum_comm\n _ = ∑ z, (∑ x, π x * (P ^ n) x z) * P z y := by\n apply Finset.sum_congr rfl; intro z _\n rw [Finset.sum_mul]; apply Finset.sum_congr rfl; intro x _; ring\n _ = ∑ z, π z * P z y := by\n apply Finset.sum_congr rfl; intro z _; rw [ih z]\n _ = π y := hstat y\n\ntheorem q222_stationary_existence_positive {Ω : Type*} [Fintype Ω] [DecidableEq Ω] [Nonempty Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (hirr : IsIrreducible P) :\n ∃ π : Ω → ℝ, IsStationary P π ∧ ∀ x, 0 < π x := by\n obtain ⟨hP0, hP1⟩ := hP\n -- Choose a starting point in the simplex.\n obtain ⟨v0, hv0⟩ : (stdSimplex ℝ Ω).Nonempty := Set.nonempty_coe_sort.mp inferInstance\n -- The iterates of `stepMap P` stay in the simplex.\n have hiter : ∀ n, (stepMap P)^[n] v0 ∈ stdSimplex ℝ Ω := by\n intro n\n induction n with\n | zero => simpa only [Function.iterate_zero_apply] using hv0\n | succ n ih => rw [Function.iterate_succ_apply']; exact stepMap_mem hP0 hP1 ih\n -- The Cesàro averages stay in the simplex.\n have hbmem : ∀ N, cesaro P v0 N ∈ stdSimplex ℝ Ω := by\n intro N\n refine ⟨fun y => ?_, ?_⟩\n · show 0 ≤ (1 / (N + 1 : ℝ)) * ∑ n ∈ Finset.range (N + 1), ((stepMap P)^[n] v0) y\n exact mul_nonneg (by positivity) (Finset.sum_nonneg (fun n _ => (hiter n).1 y))\n · show ∑ y, (1 / (N + 1 : ℝ)) * ∑ n ∈ Finset.range (N + 1), ((stepMap P)^[n] v0) y = 1\n rw [← Finset.mul_sum, Finset.sum_comm]\n rw [Finset.sum_congr rfl (fun n (_ : n ∈ Finset.range (N + 1)) => (hiter n).2)]\n rw [Finset.sum_const, Finset.card_range]\n simp only [nsmul_eq_mul, mul_one, Nat.cast_add, Nat.cast_one]\n exact one_div_mul_cancel (by positivity)\n -- Key algebraic identity: `stepMap` moves the Cesàro average by a telescoping tail.\n have hdiff : ∀ N y, stepMap P (cesaro P v0 N) y - cesaro P v0 N y\n = (1 / (N + 1 : ℝ)) * (((stepMap P)^[N + 1] v0) y - v0 y) := by\n intro N y\n have hTb : stepMap P (cesaro P v0 N) y\n = (1 / (N + 1 : ℝ)) * ∑ n ∈ Finset.range (N + 1), ((stepMap P)^[n + 1] v0) y := by\n show ∑ x, cesaro P v0 N x * P x y = _\n have e1 : ∀ x, cesaro P v0 N x * P x y\n = (1 / (N + 1 : ℝ)) * ∑ n ∈ Finset.range (N + 1), ((stepMap P)^[n] v0) x * P x y := by\n intro x\n show ((1 / (N + 1 : ℝ)) * ∑ n ∈ Finset.range (N + 1), ((stepMap P)^[n] v0) x) * P x y = _\n rw [mul_assoc, Finset.sum_mul]\n rw [Finset.sum_congr rfl (fun x (_ : x ∈ Finset.univ) => e1 x), ← Finset.mul_sum]\n congr 1\n rw [Finset.sum_comm]\n apply Finset.sum_congr rfl; intro n _\n show ∑ x, ((stepMap P)^[n] v0) x * P x y = ((stepMap P)^[n + 1] v0) y\n rw [Function.iterate_succ_apply']; rfl\n have hbNy : cesaro P v0 N y\n = (1 / (N + 1 : ℝ)) * ∑ n ∈ Finset.range (N + 1), ((stepMap P)^[n] v0) y := rfl\n rw [hTb, hbNy, ← mul_sub]\n congr 1\n rw [← Finset.sum_sub_distrib, Finset.sum_range_sub (fun n => ((stepMap P)^[n] v0) y)]\n simp [Function.iterate_zero_apply]\n -- Hence `stepMap` of the average minus the average tends to `0`.\n have htend0 : ∀ y, Filter.Tendsto (fun N => stepMap P (cesaro P v0 N) y - cesaro P v0 N y)\n Filter.atTop (nhds 0) := by\n intro y\n refine squeeze_zero_norm (fun N => ?_) tendsto_one_div_add_atTop_nhds_zero_nat\n rw [hdiff N y, norm_mul]\n have ha := mem_Icc_of_mem_stdSimplex (hiter (N + 1)) y\n have hb := mem_Icc_of_mem_stdSimplex hv0 y\n rw [Set.mem_Icc] at ha hb\n have hcnn : (0 : ℝ) ≤ 1 / ((N : ℝ) + 1) := by positivity\n rw [Real.norm_of_nonneg hcnn]\n have hbound : ‖((stepMap P)^[N + 1] v0) y - v0 y‖ ≤ 1 := by\n rw [Real.norm_eq_abs, abs_le]\n refine ⟨?_, ?_⟩ <;> linarith [ha.1, ha.2, hb.1, hb.2]\n calc 1 / ((N : ℝ) + 1) * ‖((stepMap P)^[N + 1] v0) y - v0 y‖\n ≤ 1 / ((N : ℝ) + 1) * 1 := mul_le_mul_of_nonneg_left hbound hcnn\n _ = 1 / ((N : ℝ) + 1) := mul_one _\n -- Extract a convergent subsequence of the averages; its limit is stationary.\n obtain ⟨π, hπsimplex, φ, hφmono, hφtend⟩ := (isCompact_stdSimplex ℝ Ω).tendsto_subseq hbmem\n have hπcoord : ∀ x, Filter.Tendsto (fun N => cesaro P v0 (φ N) x) Filter.atTop (nhds (π x)) :=\n fun x => tendsto_pi_nhds.mp hφtend x\n have key : ∀ y, ∑ x, π x * P x y = π y := by\n intro y\n have hL1 : Filter.Tendsto (fun N => ∑ x, cesaro P v0 (φ N) x * P x y) Filter.atTop\n (nhds (∑ x, π x * P x y)) :=\n tendsto_finset_sum _ (fun x _ => (hπcoord x).mul_const (P x y))\n have hL2 : Filter.Tendsto (fun N => cesaro P v0 (φ N) y) Filter.atTop (nhds (π y)) := hπcoord y\n have hLdiff : Filter.Tendsto\n (fun N => (∑ x, cesaro P v0 (φ N) x * P x y) - cesaro P v0 (φ N) y)\n Filter.atTop (nhds ((∑ x, π x * P x y) - π y)) := hL1.sub hL2\n have hL0 : Filter.Tendsto\n (fun N => (∑ x, cesaro P v0 (φ N) x * P x y) - cesaro P v0 (φ N) y)\n Filter.atTop (nhds 0) := (htend0 y).comp hφmono.tendsto_atTop\n have huniq := tendsto_nhds_unique hLdiff hL0\n linarith [huniq]\n refine ⟨π, ⟨hπsimplex.1, hπsimplex.2, key⟩, ?_⟩\n -- Strict positivity via irreducibility.\n intro x0\n obtain ⟨a, ha⟩ : ∃ a, 0 < π a := by\n by_contra h\n push_neg at h\n have h0 : ∑ x, π x = 0 :=\n Finset.sum_eq_zero (fun a _ => le_antisymm (h a) (hπsimplex.1 a))\n rw [hπsimplex.2] at h0\n exact one_ne_zero h0\n obtain ⟨n, _, hn2⟩ := hirr a x0\n have hpow := stationary_pow key n x0\n rw [← hpow]\n refine Finset.sum_pos'\n (fun x _ => mul_nonneg (hπsimplex.1 x) (pow_nonneg_entries hP0 n x x0)) ?_\n exact ⟨a, Finset.mem_univ a, mul_pos ha hn2⟩", "time_seconds": 194, "topic": "Markov chains (finite & countable)" }, { "question_id": 227, "problem_name": "stochastic_processes_random_walk_graph", "informal_statement": "Show that the stationary distribution for the simple random walk on the graph is deg(x)/(2|E|).", "lean_statement": "theorem q227_random_walk_graph {V : Type*} [Fintype V]\n (G : SimpleGraph V) [DecidableRel G.Adj] (hd : ∀ x, 0 < G.degree x)\n (P : Matrix V V ℝ) (hP : ∀ x y, P x y = if G.Adj x y then 1 / (G.degree x : ℝ) else 0)\n (π : V → ℝ) (hπ : ∀ x, π x = (G.degree x : ℝ) / (2 * G.edgeFinset.card)) :\n ∀ y, ∑ x, π x * P x y = π y := by sorry", "true_to_source": "literal", "proof": "theorem q227_random_walk_graph {V : Type*} [Fintype V]\n (G : SimpleGraph V) [DecidableRel G.Adj] (hd : ∀ x, 0 < G.degree x)\n (P : Matrix V V ℝ) (hP : ∀ x y, P x y = if G.Adj x y then 1 / (G.degree x : ℝ) else 0)\n (π : V → ℝ) (hπ : ∀ x, π x = (G.degree x : ℝ) / (2 * G.edgeFinset.card)) :\n ∀ y, ∑ x, π x * P x y = π y := by\n intro y\n have hstep : ∀ x, π x * P x y\n = if G.Adj x y then 1 / (2 * (G.edgeFinset.card : ℝ)) else 0 := by\n intro x\n rw [hπ, hP]\n by_cases hax : G.Adj x y\n · rw [if_pos hax, if_pos hax]\n have hdx : (G.degree x : ℝ) ≠ 0 := by exact_mod_cast (hd x).ne'\n field_simp\n · rw [if_neg hax, if_neg hax, mul_zero]\n rw [Finset.sum_congr rfl (fun x _ => hstep x)]\n rw [Finset.sum_ite, Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul]\n have hcard : (Finset.univ.filter (fun x => G.Adj x y)).card = G.degree y := by\n rw [← SimpleGraph.card_neighborFinset_eq_degree]\n congr 1\n ext x\n simp [SimpleGraph.mem_neighborFinset, G.adj_comm]\n rw [hcard, hπ y, mul_one_div]", "time_seconds": 375, "topic": "Markov chains (finite & countable)" }, { "question_id": 228, "problem_name": "stochastic_processes_kolmogorov_submartingale_inequality", "informal_statement": "Let (Z_n) be a nonnegative submartingale.\n\nShow that for every a > 0 and m ≥ 1, Pr[max_{1 ≤ i ≤ m} Z_i ≥ a] ≤ E[Z_m]/a.", "lean_statement": "theorem q228_kolmogorov_submartingale_inequality (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (Z : ℕ → Ω → ℝ) (hZ : Submartingale Z ℱ μ)\n (hnn : ∀ n ω, 0 ≤ Z n ω) (a : ℝ) (ha : 0 < a) (m : ℕ) (hm : 1 ≤ m) :\n (μ {ω | a ≤ runningMax Z m ω}).toReal ≤ (∫ ω, Z m ω ∂μ) / a := by\n sorry", "true_to_source": "literal", "proof": "theorem q228_kolmogorov_submartingale_inequality (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (Z : ℕ → Ω → ℝ) (hZ : Submartingale Z ℱ μ)\n (hnn : ∀ n ω, 0 ≤ Z n ω) (a : ℝ) (ha : 0 < a) (m : ℕ) (hm : 1 ≤ m) :\n (μ {ω | a ≤ runningMax Z m ω}).toReal ≤ (∫ ω, Z m ω ∂μ) / a := by\n have hrm : ∀ ω, runningMax Z m ω\n = (Finset.range (m+1)).sup' (by simp) (fun k => Z k ω) := by\n intro ω\n unfold runningMax\n rw [← Finset.sup'_univ_eq_ciSup]\n apply le_antisymm\n · apply Finset.sup'_le; intro k _\n apply Finset.le_sup' (fun k => Z k ω); simp [Finset.mem_range, k.isLt]\n · apply Finset.sup'_le; intro j hj\n rw [Finset.mem_range] at hj\n exact Finset.le_sup' (fun k : Fin (m+1) => Z (↑k) ω) (Finset.mem_univ ⟨j, hj⟩)\n have hcoe : ((a.toNNReal : ℝ≥0) : ℝ) = a := Real.coe_toNNReal a ha.le\n have hnn' : (0 : ℕ → Ω → ℝ) ≤ Z := fun n ω => hnn n ω\n have key := MeasureTheory.maximal_ineq hZ hnn' (ε := a.toNNReal) m\n rw [hcoe] at key\n simp only [← hrm] at key\n have hI : 0 ≤ ∫ ω in {ω | a ≤ runningMax Z m ω}, Z m ω ∂μ :=\n integral_nonneg (fun ω => hnn m ω)\n have key2 := ENNReal.toReal_mono ENNReal.ofReal_ne_top key\n rw [ENNReal.toReal_ofReal hI, ENNReal.toReal_mul, ENNReal.coe_toReal, hcoe] at key2\n -- key2 : a * (μ S).toReal ≤ ∫ ω in S, Z m ω ∂μ\n have hsub : ∫ ω in {ω | a ≤ runningMax Z m ω}, Z m ω ∂μ ≤ ∫ ω, Z m ω ∂μ :=\n setIntegral_le_integral (hZ.integrable m) (ae_of_all _ (fun ω => hnn m ω))\n have hfin : a * (μ {ω | a ≤ runningMax Z m ω}).toReal ≤ ∫ ω, Z m ω ∂μ :=\n le_trans key2 hsub\n rw [le_div_iff₀ ha, mul_comm]\n exact hfin", "time_seconds": 120, "topic": "Martingales & stopping" }, { "question_id": 230, "problem_name": "stochastic_processes_detailed_balance_stationary", "informal_statement": "If a probability distribution π satisfies the detailed balance equations π(x) P(x, y) = π(y) P(y, x) for a stochastic matrix P, then π is stationary for P.", "lean_statement": "theorem q230_detailed_balance_stationary {S : Type*} [Fintype S]\n (P : Matrix S S ℝ) (hP : IsStochastic P)\n (π : S → ℝ) (hπnn : ∀ x, 0 ≤ π x) (hπ1 : ∑ x, π x = 1)\n (hdb : DetailedBalance P π) :\n IsStationary P π := by\n sorry", "true_to_source": "abstract", "proof": "theorem q230_detailed_balance_stationary {S : Type*} [Fintype S]\n (P : Matrix S S ℝ) (hP : IsStochastic P)\n (π : S → ℝ) (hπnn : ∀ x, 0 ≤ π x) (hπ1 : ∑ x, π x = 1)\n (hdb : DetailedBalance P π) :\n IsStationary P π := by\n refine ⟨hπnn, hπ1, ?_⟩\n intro y\n have h : ∑ x, π x * P x y = ∑ x, π y * P y x := by\n apply Finset.sum_congr rfl\n intro x _\n exact hdb x y\n rw [h, ← Finset.mul_sum, hP.2 y, mul_one]", "time_seconds": 40, "topic": "Markov chains (finite & countable)" }, { "question_id": 231, "problem_name": "stochastic_processes_supermartingale_stopping_time_1", "informal_statement": "Let (X_n)_{n ≥ 0} be a supermartingale. Suppose that X is a non-negative supermartingale,\n and let T be a stopping time which is finite a.s.\n\n Show that 𝔼[X_T] ≤ 𝔼[X_0].", "lean_statement": "theorem q231_super_nonneg (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Supermartingale X ℱ μ)\n (hnn : ∀ n ω, 0 ≤ X n ω) (T : Ω → ℕ) (hT : IsStoppingTime ℱ (natStop T)) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ ≤ ∫ ω, X 0 ω ∂μ := by sorry", "true_to_source": "literal", "proof": "theorem q231_super_nonneg (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Supermartingale X ℱ μ)\n (hnn : ∀ n ω, 0 ≤ X n ω) (T : Ω → ℕ) (hT : IsStoppingTime ℱ (natStop T)) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ ≤ ∫ ω, X 0 ω ∂μ := by\n set g : Ω → ℝ := stoppedValue X (natStop T) with hg\n set F : ℕ → Ω → ℝ := fun N ω => X (min (T ω) N) ω with hF\n have hπ : ∀ N, IsStoppingTime ℱ (natStop (fun ω => min (T ω) N)) := by\n intro N; have h := hT.min_const N; convert h using 2\n have hbdd : ∀ N, ∀ ω, natStop (fun ω => min (T ω) N) ω ≤ (N : ℕ∞) := by\n intro N ω; simp only [Auto.natStop]; exact Nat.cast_le.mpr (min_le_right _ _)\n have hbound : ∀ N, ∫ ω, F N ω ∂μ ≤ ∫ ω, X 0 ω ∂μ := by\n intro N\n have hσ : IsStoppingTime ℱ (fun _ : Ω => ((0:ℕ) : ℕ∞)) := isStoppingTime_const ℱ 0\n have hle : (fun _ : Ω => ((0:ℕ) : ℕ∞)) ≤ natStop (fun ω => min (T ω) N) := by\n intro ω; simp [Auto.natStop]\n have h := hX.neg.expected_stoppedValue_mono hσ (hπ N) hle (hbdd N)\n have huntop : WithTop.untopA ((0:ℕ):ℕ∞) = 0 := rfl\n simp only [stoppedValue, Pi.neg_apply, integral_neg, huntop] at h\n have hrw : (fun a => X (natStop (fun ω => min (T ω) N) a).untopA a) = F N := by\n ext a; simp only [hF, Auto.natStop]; rfl\n rw [hrw] at h\n linarith [h]\n have hFint : ∀ N, Integrable (F N) μ := by\n intro N\n have := (hX.neg.integrable_stoppedValue (hπ N) (hbdd N)).neg\n rw [show (-stoppedValue (-X) (natStop (fun ω => min (T ω) N))) = F N by\n ext a; simp only [hF, stoppedValue, Pi.neg_apply, Auto.natStop, neg_neg]; rfl] at this\n exact this\n have hFnn : ∀ N ω, 0 ≤ F N ω := fun N ω => hnn _ _\n have hgnn : ∀ ω, 0 ≤ g ω := by\n intro ω; simp only [hg, stoppedValue]; exact hnn _ _\n have h_lim : ∀ ω, Tendsto (fun N => F N ω) atTop (nhds (g ω)) := by\n intro ω\n have hv : g ω = X (T ω) ω := by simp only [hg, stoppedValue, Auto.natStop]; rfl\n rw [hv]\n apply tendsto_atTop_of_eventually_const (i₀ := T ω)\n intro N hN; simp only [hF]; rw [min_eq_left hN]\n have hgmeas : AEStronglyMeasurable g μ := by\n apply AEMeasurable.aestronglyMeasurable\n apply aemeasurable_of_tendsto_metrizable_ae atTop\n (fun N => (hFint N).aemeasurable)\n filter_upwards with ω using h_lim ω\n have hofR : ∀ N, ∫⁻ ω, ENNReal.ofReal (F N ω) ∂μ = ENNReal.ofReal (∫ ω, F N ω ∂μ) := by\n intro N\n rw [← ofReal_integral_eq_lintegral_ofReal (hFint N) (Filter.Eventually.of_forall (hFnn N))]\n have hX0nn : 0 ≤ ∫ ω, X 0 ω ∂μ := integral_nonneg (fun ω => hnn 0 ω)\n have hfatou : ∫⁻ ω, ENNReal.ofReal (g ω) ∂μ ≤ ENNReal.ofReal (∫ ω, X 0 ω ∂μ) := by\n have hli : ∫⁻ ω, ENNReal.ofReal (g ω) ∂μ\n ≤ liminf (fun N => ∫⁻ ω, ENNReal.ofReal (F N ω) ∂μ) atTop := by\n have hlimeq : (fun ω => ENNReal.ofReal (g ω))\n = fun ω => liminf (fun N => ENNReal.ofReal (F N ω)) atTop := by\n ext ω\n have ht : Tendsto (fun N => ENNReal.ofReal (F N ω)) atTop (nhds (ENNReal.ofReal (g ω))) :=\n (ENNReal.continuous_ofReal.tendsto _).comp (h_lim ω)\n exact (ht.liminf_eq).symm\n rw [hlimeq]\n exact lintegral_liminf_le' (fun N => (hFint N).aemeasurable.ennreal_ofReal)\n refine hli.trans ?_\n rw [show (fun N => ∫⁻ ω, ENNReal.ofReal (F N ω) ∂μ)\n = fun N => ENNReal.ofReal (∫ ω, F N ω ∂μ) from funext hofR]\n apply liminf_le_of_le\n · exact ⟨0, by simp⟩\n · intro b hb\n obtain ⟨N, hN⟩ := hb.exists\n exact le_trans hN (ENNReal.ofReal_le_ofReal (hbound N))\n rw [integral_eq_lintegral_of_nonneg_ae (Filter.Eventually.of_forall hgnn) hgmeas]\n rw [← ENNReal.toReal_ofReal hX0nn]\n exact ENNReal.toReal_le_toReal (ne_of_lt (lt_of_le_of_lt hfatou ENNReal.ofReal_lt_top))\n ENNReal.ofReal_ne_top |>.mpr hfatou", "time_seconds": 85, "topic": "Martingales & stopping" }, { "question_id": 234, "problem_name": "stochastic_processes_dependent_zero_mean_martingale", "informal_statement": "Let (X_i) be random variables with E[|X_i|] < ∞ and E[X_i | X_{i−1}, …, X_1] = 0 for all i ≥ 2, with E[X_1] = 0, and let Z_n = X_1 + … + X_n.\n\nShow that (Z_n) is a zero-mean martingale.", "lean_statement": "theorem q234_mds_martingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hadap : Adapted ℱ X) (hint : ∀ n, Integrable (X n) μ)\n (hmean0 : ∫ ω, X 0 ω ∂μ = 0)\n (hmds : ∀ n, μ[X (n + 1) | ℱ n] =ᵐ[μ] 0) :\n Martingale (fun n ω => ∑ i ∈ Finset.range (n + 1), X i ω) ℱ μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q234_mds_martingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hadap : Adapted ℱ X) (hint : ∀ n, Integrable (X n) μ)\n (hmean0 : ∫ ω, X 0 ω ∂μ = 0)\n (hmds : ∀ n, μ[X (n + 1) | ℱ n] =ᵐ[μ] 0) :\n Martingale (fun n ω => ∑ i ∈ Finset.range (n + 1), X i ω) ℱ μ := by\n have hsX : StronglyAdapted ℱ X := hadap.stronglyAdapted\n refine martingale_of_condExp_sub_eq_zero_nat ?_ ?_ ?_\n · intro n\n refine Finset.stronglyMeasurable_fun_sum _ (fun i hi => ?_)\n exact (hsX i).mono (ℱ.mono (Nat.lt_succ_iff.mp (Finset.mem_range.mp hi)))\n · intro n\n exact integrable_finsetSum _ (fun i _ => hint i)\n · intro i\n have hcong : (fun ω => ∑ j ∈ Finset.range (i + 1 + 1), X j ω)\n - (fun ω => ∑ j ∈ Finset.range (i + 1), X j ω) =ᵐ[μ] X (i + 1) := by\n filter_upwards with ω\n simp only [Pi.sub_apply, Finset.sum_range_succ]\n ring\n exact (condExp_congr_ae hcong).trans (hmds i)", "time_seconds": 472, "topic": "Martingales & stopping" }, { "question_id": 244, "problem_name": "stochastic_processes_submartingale_2", "informal_statement": "Consider biased gambler’s ruin: at each step, the gambler gains one dollar\n with probability p and losses one dollar with probability (1 − p).\n Let X_n be the money in purse at time n.\n\n Show that if p > 1/2, then (X_n) is a submartingale.", "lean_statement": "theorem q244_gambler_submartingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hadap : Adapted ℱ X) (hint : ∀ n, Integrable (X n) μ) (p : ℝ) (hp : 1 / 2 < p)\n (hstep : IsConstDrift μ ℱ X (2 * p - 1)) :\n Submartingale X ℱ μ := by sorry", "true_to_source": "literal", "proof": "theorem q244_gambler_submartingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hadap : Adapted ℱ X) (hint : ∀ n, Integrable (X n) μ) (p : ℝ) (hp : 1 / 2 < p)\n (hstep : IsConstDrift μ ℱ X (2 * p - 1)) :\n Submartingale X ℱ μ := by\n apply submartingale_of_condExp_sub_nonneg_nat hadap.stronglyAdapted hint\n intro i\n refine (Filter.EventuallyLE.trans_eq ?_ (hstep i).symm)\n refine ae_of_all μ (fun ω => ?_)\n simp only [Pi.zero_apply]\n linarith", "time_seconds": 60, "topic": "Martingales & stopping" }, { "question_id": 245, "problem_name": "stochastic_processes_martingale_partial_isolation", "informal_statement": "Let (Z_n) be a martingale.\n\nShow that for any n > i ≥ 1, E[Z_n | Z_i, Z_{i−1}, …, Z_1] = Z_i, and consequently E[Z_n] = E[Z_1].", "lean_statement": "theorem q245_martingale_tower (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (Z : ℕ → Ω → ℝ) (hZ : Martingale Z ℱ μ) (i n : ℕ) (hin : i ≤ n) :\n μ[Z n | ℱ i] =ᵐ[μ] Z i ∧ ∫ ω, Z n ω ∂μ = ∫ ω, Z i ω ∂μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q245_martingale_tower (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (Z : ℕ → Ω → ℝ) (hZ : Martingale Z ℱ μ) (i n : ℕ) (hin : i ≤ n) :\n μ[Z n | ℱ i] =ᵐ[μ] Z i ∧ ∫ ω, Z n ω ∂μ = ∫ ω, Z i ω ∂μ := by\n refine ⟨hZ.condExp_ae_eq hin, ?_⟩\n have h1 : ∫ ω, (μ[Z n | ℱ i]) ω ∂μ = ∫ ω, Z n ω ∂μ := integral_condExp (ℱ.le i)\n rw [← h1]\n exact integral_congr_ae (hZ.condExp_ae_eq hin)", "time_seconds": 46, "topic": "Martingales & stopping" }, { "question_id": 248, "problem_name": "stochastic_processes_sprt_expected_sample_size", "informal_statement": "Consider a sequential probability ratio test with thresholds α > 0 and β < 0, stopping at trial J, where the per-observation log-likelihood ratio Z has E[Z | H = 0] > 0.\n\nShow that E[J | H = 0] = (α Pr[cross α | H=0] + β Pr[cross β | H=0]) / E[Z | H = 0] by Wald's equality.", "lean_statement": "theorem q248_sprt_expected_sample_size (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ]\n (Z S M : ℕ → Ω → ℝ) (d : ℝ) (hd : 0 < d)\n (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, Z i ω)\n (hmean : ∫ ω, Z 0 ω ∂μ = d)\n (hM : ∀ n ω, M n ω = S n ω - (n : ℝ) * d)\n (hMmart : Martingale M ℱ μ)\n (J : Ω → ℕ) (hJ : IsStoppingTime ℱ (natStop J)) (N : ℕ) (hJbdd : ∀ ω, J ω ≤ N)\n (hEJ : Integrable (fun ω => (J ω : ℝ)) μ)\n (α β : ℝ) (hα : 0 < α) (hβ : β < 0)\n (hhit : ∀ ω, S (J ω) ω = α ∨ S (J ω) ω = β) :\n ∫ ω, (J ω : ℝ) ∂μ\n = (α * (μ {ω | S (J ω) ω = α}).toReal + β * (μ {ω | S (J ω) ω = β}).toReal) / d := by\n sorry", "true_to_source": "abstract", "proof": "theorem q248_sprt_expected_sample_size (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ]\n (Z S M : ℕ → Ω → ℝ) (d : ℝ) (hd : 0 < d)\n (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, Z i ω)\n (hmean : ∫ ω, Z 0 ω ∂μ = d)\n (hM : ∀ n ω, M n ω = S n ω - (n : ℝ) * d)\n (hMmart : Martingale M ℱ μ)\n (J : Ω → ℕ) (hJ : IsStoppingTime ℱ (natStop J)) (N : ℕ) (hJbdd : ∀ ω, J ω ≤ N)\n (hEJ : Integrable (fun ω => (J ω : ℝ)) μ)\n (α β : ℝ) (hα : 0 < α) (hβ : β < 0)\n (hhit : ∀ ω, S (J ω) ω = α ∨ S (J ω) ω = β) :\n ∫ ω, (J ω : ℝ) ∂μ\n = (α * (μ {ω | S (J ω) ω = α}).toReal + β * (μ {ω | S (J ω) ω = β}).toReal) / d := by\n have hle : ∀ ω, natStop J ω ≤ (N : WithTop ℕ) := by\n intro ω; unfold natStop; exact_mod_cast hJbdd ω\n -- M 0 = 0\n have hM0 : ∫ ω, M 0 ω ∂μ = 0 := by\n have h0 : ∀ ω, M 0 ω = 0 := by intro ω; rw [hM 0 ω, hS 0 ω]; simp\n simp only [h0, integral_zero]\n -- E[stoppedValue] = E[M N] = E[M 0] = 0\n have hcond := hMmart.stoppedValue_ae_eq_condExp_of_le_const hJ hle (n := N)\n have hMN0 : ∫ ω, M N ω ∂μ = ∫ ω, M 0 ω ∂μ := by\n have := hMmart.setIntegral_eq (i := 0) (j := N) (Nat.zero_le N) (s := Set.univ)\n MeasurableSet.univ\n simpa using this.symm\n have heq0 : ∫ ω, M (J ω) ω ∂μ = 0 := by\n have hsv : (fun ω => M (J ω) ω) = stoppedValue M (natStop J) := by\n unfold stoppedValue natStop; rfl\n rw [hsv, integral_congr_ae hcond, integral_condExp hJ.measurableSpace_le, hMN0, hM0]\n -- integrability of M (J ·)\n have hMJ_int : Integrable (fun ω => M (J ω) ω) μ := by\n have hsv : (fun ω => M (J ω) ω) = stoppedValue M (natStop J) := by\n unfold stoppedValue natStop; rfl\n rw [hsv]\n have hmemlp : MemLp (stoppedValue M (natStop J)) 1 μ :=\n memLp_stoppedValue hJ (fun n => (memLp_one_iff_integrable).mpr (hMmart.integrable n)) hle\n exact (memLp_one_iff_integrable).mp hmemlp\n have hJd_int : Integrable (fun ω => (J ω : ℝ) * d) μ := hEJ.mul_const d\n -- S (J ·) = M (J ·) + J·d\n have hSeq : (fun ω => S (J ω) ω) = (fun ω => M (J ω) ω) + (fun ω => (J ω : ℝ) * d) := by\n funext ω; simp only [Pi.add_apply]; have := hM (J ω) ω; linarith\n have hSJval : ∫ ω, S (J ω) ω ∂μ = (∫ ω, (J ω : ℝ) ∂μ) * d := by\n have hcongr : ∫ ω, S (J ω) ω ∂μ = ∫ ω, (M (J ω) ω + (J ω : ℝ) * d) ∂μ := by\n apply integral_congr_ae; filter_upwards with ω; have := hM (J ω) ω; linarith\n rw [hcongr, integral_add hMJ_int hJd_int, heq0, zero_add, integral_mul_const]\n -- measurability of g = S (J ·)\n have hSadapt : StronglyAdapted ℱ S := by\n intro i\n have hSi : S i = fun ω => M i ω + (i : ℝ) * d := by funext ω; rw [hM i ω]; ring\n rw [hSi]\n exact (hMmart.1 i).add stronglyMeasurable_const\n have hSprog : ProgMeasurable ℱ S := hSadapt.progMeasurable_of_discrete\n have hgmeas : StronglyMeasurable (fun ω => S (J ω) ω) := by\n have hsv : (fun ω => S (J ω) ω) = stoppedValue S (natStop J) := by\n unfold stoppedValue natStop; rfl\n rw [hsv]\n exact (stronglyMeasurable_stoppedValue_of_le hSprog hJ hle).mono (ℱ.le N)\n have hαβ : α ≠ β := (lt_trans hβ hα).ne'\n have hmsα : MeasurableSet {ω | S (J ω) ω = α} := hgmeas.measurable (measurableSet_singleton α)\n have hmsβ : MeasurableSet {ω | S (J ω) ω = β} := hgmeas.measurable (measurableSet_singleton β)\n -- value split\n have hgeq : (fun ω => S (J ω) ω)\n = ({ω | S (J ω) ω = α}.indicator (fun _ => α))\n + ({ω | S (J ω) ω = β}.indicator (fun _ => β)) := by\n funext ω\n simp only [Pi.add_apply]\n rcases hhit ω with h | h\n · rw [Set.indicator_of_mem (by exact h), Set.indicator_of_notMem (by simp [h, hαβ])]\n simp [h]\n · rw [Set.indicator_of_notMem (by simp [h, hαβ.symm]), Set.indicator_of_mem (by exact h)]\n simp [h]\n have hintα : Integrable ({ω | S (J ω) ω = α}.indicator (fun _ => α)) μ :=\n (integrable_const α).indicator hmsα\n have hintβ : Integrable ({ω | S (J ω) ω = β}.indicator (fun _ => β)) μ :=\n (integrable_const β).indicator hmsβ\n have hsplit : ∫ ω, S (J ω) ω ∂μ\n = α * (μ {ω | S (J ω) ω = α}).toReal + β * (μ {ω | S (J ω) ω = β}).toReal := by\n have hc : ∫ ω, S (J ω) ω ∂μ\n = ∫ ω, ({ω | S (J ω) ω = α}.indicator (fun _ => α) ω\n + {ω | S (J ω) ω = β}.indicator (fun _ => β) ω) ∂μ := by\n apply integral_congr_ae\n filter_upwards with ω\n have h2 := congrFun hgeq ω\n simp only [Pi.add_apply] at h2\n exact h2\n rw [hc, integral_add hintα hintβ, integral_indicator_const _ hmsα,\n integral_indicator_const _ hmsβ]\n simp only [smul_eq_mul, MeasureTheory.measureReal_def]\n ring\n -- combine\n rw [eq_div_iff (ne_of_gt hd), ← hsplit, hSJval]", "time_seconds": 815, "topic": "Random walks & large deviations" }, { "question_id": 251, "problem_name": "stochastic_processes_random_walk_recurrent_2", "informal_statement": "Show that simple random walk on ℤ^1 is recurrent.", "lean_statement": "theorem q251_z1_recurrent :\n ¬ Summable (fun n : ℕ => (Nat.choose (2 * n) n : ℝ) / (4 : ℝ) ^ n) := by sorry", "true_to_source": "abstract", "proof": "theorem q251_z1_recurrent :\n ¬ Summable (fun n : ℕ => (Nat.choose (2 * n) n : ℝ) / (4 : ℝ) ^ n) := by\n intro hs\n have key : Summable (fun n : ℕ => (1/2) * (1/((n:ℝ)+1))) := by\n apply Summable.of_nonneg_of_le (g := fun n : ℕ => (1/2) * (1/((n:ℝ)+1)))\n (f := fun n : ℕ => (Nat.choose (2 * n) n : ℝ) / (4 : ℝ) ^ n)\n (fun n => by positivity) ?_ hs\n intro n\n have hc : (0:ℝ) ≤ (Nat.choose (2*n) n : ℝ) := by positivity\n have hb : (4:ℝ)^n ≤ (2*(n:ℝ)+1) * (Nat.choose (2*n) n : ℝ) := by\n have h := Nat.four_pow_le_two_mul_add_one_mul_central_binom n\n have h2 := (Nat.cast_le (α:=ℝ)).mpr h\n push_cast at h2\n linarith [h2]\n have h4 : (0:ℝ) < (4:ℝ)^n := by positivity\n have hd : (0:ℝ) < 2*((n:ℝ)+1) := by positivity\n have goal2 : 1/(2*((n:ℝ)+1)) ≤ (Nat.choose (2*n) n : ℝ)/(4:ℝ)^n := by\n rw [div_le_div_iff₀ hd h4]\n nlinarith [hb, hc]\n calc (1/2)*(1/((n:ℝ)+1)) = 1/(2*((n:ℝ)+1)) := by rw [one_div_mul_one_div]\n _ ≤ _ := goal2\n have key2 : Summable (fun n : ℕ => (1:ℝ)/((n:ℝ)+1)) := by\n have h := key.mul_left 2\n have heq : (fun n : ℕ => 2 * ((1/2) * (1/((n:ℝ)+1)))) = (fun n : ℕ => (1:ℝ)/((n:ℝ)+1)) := by\n funext n; ring\n rwa [heq] at h\n have key3 : Summable (fun n : ℕ => (1:ℝ)/(n:ℝ)) := by\n rw [← summable_nat_add_iff 1]\n simpa using key2\n exact Real.not_summable_one_div_natCast key3", "time_seconds": 119, "topic": "Random walks & large deviations" }, { "question_id": 252, "problem_name": "stochastic_processes_birth_death_balance_equation", "informal_statement": "Consider an irreducible positive-recurrent birth-death Markov chain on {0,1,2,…} with stationary distribution π, up-probability p_i and down-probability q_{i+1}.\n\nShow that the stationary distribution satisfies the detailed balance relation π_i p_i = π_{i+1} q_{i+1} for all i ≥ 0.", "lean_statement": "theorem q252_birth_death_detailed_balance\n (p q π : ℕ → ℝ)\n (hp : ∀ i, 0 ≤ p i) (hq : ∀ i, 0 ≤ q i) (hπ : ∀ i, 0 ≤ π i)\n (hbal0 : π 0 * p 0 = π 1 * q 1)\n (hbal : ∀ k, π (k + 1) * (p (k + 1) + q (k + 1)) = π k * p k + π (k + 2) * q (k + 2)) :\n ∀ i, π i * p i = π (i + 1) * q (i + 1) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q252_birth_death_detailed_balance\n (p q π : ℕ → ℝ)\n (hp : ∀ i, 0 ≤ p i) (hq : ∀ i, 0 ≤ q i) (hπ : ∀ i, 0 ≤ π i)\n (hbal0 : π 0 * p 0 = π 1 * q 1)\n (hbal : ∀ k, π (k + 1) * (p (k + 1) + q (k + 1)) = π k * p k + π (k + 2) * q (k + 2)) :\n ∀ i, π i * p i = π (i + 1) * q (i + 1) := by\n intro i\n induction i with\n | zero => exact hbal0\n | succ k ih =>\n have h := hbal k\n rw [mul_add] at h\n linarith [ih, h]", "time_seconds": 47, "topic": "Markov chains (finite & countable)" }, { "question_id": 257, "problem_name": "stochastic_processes_stopped_martingale_expectation_equal", "informal_statement": "Let (Z_n) be a martingale and J a stopping trial with stopped process (Z_n*).\n\nShow that E[Z_1] = E[Z_n*] = E[Z_n] for all n.", "lean_statement": "theorem q257_stopped_martingale_expectation_equal (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ]\n (Z : ℕ → Ω → ℝ) (hZ : Martingale Z ℱ μ)\n (J : Ω → ℕ) (hJ : IsStoppingTime ℱ (natStop J)) (n : ℕ) :\n ∫ ω, Z 0 ω ∂μ = ∫ ω, stoppedProcess Z (natStop J) n ω ∂μ ∧\n ∫ ω, stoppedProcess Z (natStop J) n ω ∂μ = ∫ ω, Z n ω ∂μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q257_stopped_martingale_expectation_equal (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ]\n (Z : ℕ → Ω → ℝ) (hZ : Martingale Z ℱ μ)\n (J : Ω → ℕ) (hJ : IsStoppingTime ℱ (natStop J)) (n : ℕ) :\n ∫ ω, Z 0 ω ∂μ = ∫ ω, stoppedProcess Z (natStop J) n ω ∂μ ∧\n ∫ ω, stoppedProcess Z (natStop J) n ω ∂μ = ∫ ω, Z n ω ∂μ := by\n have hsub : Submartingale (stoppedProcess Z (natStop J)) ℱ μ := hZ.submartingale.stoppedProcess hJ\n have hsuper : Supermartingale (stoppedProcess Z (natStop J)) ℱ μ := by\n have h1 : Submartingale (stoppedProcess (-Z) (natStop J)) ℱ μ :=\n hZ.supermartingale.neg.stoppedProcess hJ\n have h3 : -(stoppedProcess (-Z) (natStop J)) = stoppedProcess Z (natStop J) := by\n funext i ω; simp [MeasureTheory.stoppedProcess]\n have h2 := h1.neg\n rwa [h3] at h2\n have hmart : Martingale (stoppedProcess Z (natStop J)) ℱ μ :=\n martingale_iff.mpr ⟨hsuper, hsub⟩\n have hsp0 : stoppedProcess Z (natStop J) 0 = Z 0 := by\n funext ω; simp [MeasureTheory.stoppedProcess]\n have h1 := hmart.setIntegral_eq (Nat.zero_le n) (s := Set.univ) MeasurableSet.univ\n have h2 := hZ.setIntegral_eq (Nat.zero_le n) (s := Set.univ) MeasurableSet.univ\n rw [setIntegral_univ, setIntegral_univ, hsp0] at h1\n rw [setIntegral_univ, setIntegral_univ] at h2\n exact ⟨h1, by rw [← h1, h2]⟩", "time_seconds": 157, "topic": "Martingales & stopping" }, { "question_id": 259, "problem_name": "stochastic_processes_martingale_stopping_time_4", "informal_statement": "Let (X_n)_{n ≥ 0} be a martingale, and let T be a stopping time with 𝔼[T] < ∞.\n You are given that X has bounded increments, i.e. ∃M > 0 such that |X_{n+1} − X_n| ≤ M for all n.\n\n Show that 𝔼[X_T] = 𝔼[X_0].", "lean_statement": "theorem q259_optional_stopping_bdd_incr (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Martingale X ℱ μ) (T : Ω → ℕ)\n (hT : IsStoppingTime ℱ (natStop T)) (hET : Integrable (fun ω => (T ω : ℝ)) μ)\n (M : ℝ) (hincr : ∀ n, ∀ᵐ ω ∂μ, |X (n + 1) ω - X n ω| ≤ M) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ = ∫ ω, X 0 ω ∂μ := by sorry", "true_to_source": "literal", "proof": "theorem q259_optional_stopping_bdd_incr (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Martingale X ℱ μ) (T : Ω → ℕ)\n (hT : IsStoppingTime ℱ (natStop T)) (hET : Integrable (fun ω => (T ω : ℝ)) μ)\n (M : ℝ) (hincr : ∀ n, ∀ᵐ ω ∂μ, |X (n + 1) ω - X n ω| ≤ M) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ = ∫ ω, X 0 ω ∂μ := by\n classical\n change ∫ ω, X (T ω) ω ∂μ = ∫ ω, X 0 ω ∂μ\n -- 0 ≤ M\n have hM : 0 ≤ M := by\n obtain ⟨ω, hω⟩ := (hincr 0).exists\n exact le_trans (abs_nonneg _) hω\n -- telescoping bound\n have htel : ∀ (ω : Ω), (∀ i, |X (i + 1) ω - X i ω| ≤ M) →\n ∀ k, |X k ω| ≤ |X 0 ω| + M * (k : ℝ) := by\n intro ω hω k\n induction k with\n | zero => simp\n | succ j ih =>\n have hstep : |X (j + 1) ω| ≤ |X j ω| + |X (j + 1) ω - X j ω| := by\n calc |X (j + 1) ω| = |X j ω + (X (j + 1) ω - X j ω)| := by ring_nf\n _ ≤ |X j ω| + |X (j + 1) ω - X j ω| := abs_add_le _ _\n have hb := hω j\n push_cast\n nlinarith [ih, hb, hstep]\n -- ae version of increments\n have haeincr : ∀ᵐ ω ∂μ, ∀ i, |X (i + 1) ω - X i ω| ≤ M := ae_all_iff.mpr hincr\n -- truncated stopping times\n have hσ_st : ∀ n : ℕ, IsStoppingTime ℱ (fun ω => min (natStop T ω) (n : ℕ∞)) :=\n fun n => hT.min_const n\n have hσ_bdd : ∀ (n : ℕ) (ω : Ω), (fun ω => min (natStop T ω) (n : ℕ∞)) ω ≤ (n : ℕ∞) :=\n fun n ω => min_le_right _ _\n -- reduction of stoppedValue at truncated time\n have hFred : ∀ (n : ℕ) (ω : Ω),\n stoppedValue X (fun ω => min (natStop T ω) (n : ℕ∞)) ω = X (min (T ω) n) ω := by\n intro n ω\n have h : (min (natStop T ω) (n : ℕ∞)).untopA = min (T ω) n := by\n simp only [natStop]; norm_cast\n simp only [stoppedValue, h]\n -- bounded OST: constant expectation of stopped value\n have hstop_eq : ∀ n : ℕ,\n ∫ ω, stoppedValue X (fun ω => min (natStop T ω) (n : ℕ∞)) ω ∂μ = ∫ ω, X 0 ω ∂μ := by\n intro n\n refine le_antisymm ?_ ?_\n · have h2 := hX.neg.submartingale.expected_stoppedValue_mono\n (isStoppingTime_const ℱ 0) (hσ_st n) (fun ω => zero_le') (N := n) (hσ_bdd n)\n rw [stoppedValue_const] at h2\n have e1 : ∫ ω, stoppedValue (-X) (fun ω => min (natStop T ω) (n : ℕ∞)) ω ∂μ\n = -∫ ω, stoppedValue X (fun ω => min (natStop T ω) (n : ℕ∞)) ω ∂μ := by\n rw [← integral_neg]; rfl\n have e2 : ∫ ω, (-X) 0 ω ∂μ = -∫ ω, X 0 ω ∂μ := by\n rw [← integral_neg]; rfl\n rw [e1, e2] at h2\n linarith\n · have h1 := hX.submartingale.expected_stoppedValue_mono\n (isStoppingTime_const ℱ 0) (hσ_st n) (fun ω => zero_le') (N := n) (hσ_bdd n)\n rw [stoppedValue_const] at h1\n linarith\n -- dominated convergence\n have hg_int : Integrable (fun ω => |X 0 ω| + M * (T ω : ℝ)) μ :=\n ((hX.integrable 0).abs).add (hET.const_mul M)\n have hF_meas : ∀ n : ℕ, AEStronglyMeasurable\n (stoppedValue X (fun ω => min (natStop T ω) (n : ℕ∞))) μ :=\n fun n => (hX.submartingale.integrable_stoppedValue (hσ_st n) (hσ_bdd n)).aestronglyMeasurable\n have h_bound : ∀ n : ℕ, ∀ᵐ ω ∂μ,\n ‖stoppedValue X (fun ω => min (natStop T ω) (n : ℕ∞)) ω‖ ≤ |X 0 ω| + M * (T ω : ℝ) := by\n intro n\n filter_upwards [haeincr] with ω hω\n rw [Real.norm_eq_abs, hFred n ω]\n have hk := htel ω hω (min (T ω) n)\n have hmin : ((min (T ω) n : ℕ) : ℝ) ≤ (T ω : ℝ) := by exact_mod_cast Nat.min_le_left _ _\n nlinarith [hk, hmin, hM, abs_nonneg (X 0 ω)]\n have h_lim : ∀ᵐ ω ∂μ, Tendsto\n (fun n : ℕ => stoppedValue X (fun ω => min (natStop T ω) (n : ℕ∞)) ω) atTop\n (𝓝 (X (T ω) ω)) := by\n apply ae_of_all\n intro ω\n apply tendsto_atTop_of_eventually_const (i₀ := T ω)\n intro n hn\n rw [hFred n ω, min_eq_left hn]\n have hdct := tendsto_integral_of_dominated_convergence\n (fun ω => |X 0 ω| + M * (T ω : ℝ)) hF_meas hg_int h_bound h_lim\n have hconst : Tendsto\n (fun n : ℕ => ∫ ω, stoppedValue X (fun ω => min (natStop T ω) (n : ℕ∞)) ω ∂μ) atTop\n (𝓝 (∫ ω, X 0 ω ∂μ)) := by\n simp only [hstop_eq]\n exact tendsto_const_nhds\n exact tendsto_nhds_unique hdct hconst", "time_seconds": 560, "topic": "Martingales & stopping" }, { "question_id": 262, "problem_name": "stochastic_processes_nonneg_supermartingale_convergence", "informal_statement": "Let (Z_n) be a nonnegative supermartingale.\n\nShow that Z_n converges with probability 1 to a finite limiting random variable as n → ∞.", "lean_statement": "theorem q262_nonneg_supermartingale_convergence (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (Z : ℕ → Ω → ℝ) (hZ : Supermartingale Z ℱ μ)\n (hnn : ∀ n ω, 0 ≤ Z n ω) :\n ∃ Zinf : Ω → ℝ, ∀ᵐ ω ∂μ, Tendsto (fun n => Z n ω) atTop (𝓝 (Zinf ω)) := by\n sorry", "true_to_source": "literal", "proof": "theorem q262_nonneg_supermartingale_convergence (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (Z : ℕ → Ω → ℝ) (hZ : Supermartingale Z ℱ μ)\n (hnn : ∀ n ω, 0 ≤ Z n ω) :\n ∃ Zinf : Ω → ℝ, ∀ᵐ ω ∂μ, Tendsto (fun n => Z n ω) atTop (𝓝 (Zinf ω)) := by\n have hsub : Submartingale (-Z) ℱ μ := hZ.neg\n have hI : ∀ n, (∫ ω, Z n ω ∂μ) ≤ ∫ ω, Z 0 ω ∂μ := by\n intro n\n calc ∫ ω, Z n ω ∂μ\n = ∫ ω, (μ[Z n | ℱ 0]) ω ∂μ := (integral_condExp (ℱ.le 0)).symm\n _ ≤ ∫ ω, Z 0 ω ∂μ :=\n integral_mono_ae integrable_condExp (hZ.integrable 0) (hZ.condExp_ae_le (Nat.zero_le n))\n have hbdd : ∀ n, eLpNorm ((-Z) n) 1 μ ≤ ENNReal.ofReal (∫ ω, Z 0 ω ∂μ) := by\n intro n\n have hZneq : ((-Z) n) = -(Z n) := rfl\n rw [hZneq, eLpNorm_neg, eLpNorm_one_eq_lintegral_enorm]\n have hpt : ∀ ω, ‖Z n ω‖ₑ = ENNReal.ofReal (Z n ω) := fun ω => Real.enorm_eq_ofReal (hnn n ω)\n calc (∫⁻ ω, ‖Z n ω‖ₑ ∂μ)\n = ∫⁻ ω, ENNReal.ofReal (Z n ω) ∂μ := by simp_rw [hpt]\n _ = ENNReal.ofReal (∫ ω, Z n ω ∂μ) :=\n (ofReal_integral_eq_lintegral_ofReal (hZ.integrable n) (ae_of_all μ (hnn n))).symm\n _ ≤ ENNReal.ofReal (∫ ω, Z 0 ω ∂μ) := ENNReal.ofReal_le_ofReal (hI n)\n refine ⟨fun ω => -(ℱ.limitProcess (-Z) μ ω), ?_⟩\n filter_upwards [hsub.ae_tendsto_limitProcess hbdd] with ω hω\n have hneg := hω.neg\n simpa using hneg", "time_seconds": 225, "topic": "Martingales & stopping" }, { "question_id": 265, "problem_name": "stochastic_processes_product_form_exponential_martingale", "informal_statement": "Let (S_n) be a random walk with i.i.d. steps X whose semi-invariant generating function γ(r) = ln E[e^{rX}] exists near r, and let Z_n = exp(r S_n − n γ(r)).\n\nShow that (Z_n) is a martingale.", "lean_statement": "theorem q265_exponential_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hmeas : ∀ i, StronglyMeasurable (X i))\n (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (r γ : ℝ) (hγ : ∫ ω, Real.exp (r * X 0 ω) ∂μ = Real.exp γ)\n (hint : ∀ n, Integrable\n (fun ω => Real.exp (r * (∑ i ∈ Finset.range (n + 1), X i ω) - ((n : ℝ) + 1) * γ)) μ) :\n Martingale\n (fun n ω => Real.exp (r * (∑ i ∈ Finset.range (n + 1), X i ω) - ((n : ℝ) + 1) * γ))\n (Filtration.natural X hmeas) μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q265_exponential_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hmeas : ∀ i, StronglyMeasurable (X i))\n (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (r γ : ℝ) (hγ : ∫ ω, Real.exp (r * X 0 ω) ∂μ = Real.exp γ)\n (hint : ∀ n, Integrable\n (fun ω => Real.exp (r * (∑ i ∈ Finset.range (n + 1), X i ω) - ((n : ℝ) + 1) * γ)) μ) :\n Martingale\n (fun n ω => Real.exp (r * (∑ i ∈ Finset.range (n + 1), X i ω) - ((n : ℝ) + 1) * γ))\n (Filtration.natural X hmeas) μ := by\n set ℱ := Filtration.natural X hmeas with hℱ\n have hadapt : StronglyAdapted ℱ X := Filtration.stronglyAdapted_natural hmeas\n have hXmeas : ∀ i k, k ≤ i → StronglyMeasurable[ℱ i] (X k) := by\n intro i k hk\n exact (hadapt k).mono (ℱ.mono hk)\n have hcont : Continuous fun x : ℝ => Real.exp (r * x - γ) := by fun_prop\n have hZeq : ∀ n : ℕ,\n (fun ω => Real.exp (r * (∑ k ∈ Finset.range (n + 1), X k ω) - ((n : ℝ) + 1) * γ))\n = fun ω => ∏ k ∈ Finset.range (n + 1), Real.exp (r * X k ω - γ) := by\n intro n; funext ω\n rw [← Real.exp_sum]\n congr 1\n rw [Finset.sum_sub_distrib, ← Finset.mul_sum, Finset.sum_const, Finset.card_range,\n nsmul_eq_mul]\n push_cast; ring\n have hprodfun : ∀ m : ℕ, (fun ω => ∏ k ∈ Finset.range (m + 1), Real.exp (r * X k ω - γ))\n = ∏ k ∈ Finset.range (m + 1), (fun ω => Real.exp (r * X k ω - γ)) :=\n fun m => by funext ω; rw [Finset.prod_apply]\n have hAsm : ∀ i, StronglyMeasurable[ℱ i]\n (fun ω => ∏ k ∈ Finset.range (i + 1), Real.exp (r * X k ω - γ)) := by\n intro i\n rw [hprodfun i]\n exact Finset.stronglyMeasurable_prod _\n (fun k hk => hcont.comp_stronglyMeasurable (hXmeas i k (Nat.lt_succ_iff.mp (Finset.mem_range.mp hk))))\n have hint' : ∀ n, Integrable\n (fun ω => ∏ k ∈ Finset.range (n + 1), Real.exp (r * X k ω - γ)) μ :=\n fun n => by rw [← hZeq n]; exact hint n\n have hYint1 : ∀ k, ∫ ω, Real.exp (r * X k ω - γ) ∂μ = 1 := by\n intro k\n have hid : IdentDistrib (fun ω => Real.exp (r * X k ω - γ))\n (fun ω => Real.exp (r * X 0 ω - γ)) μ μ := (hident k).comp hcont.measurable\n rw [hid.integral_eq]\n simp_rw [Real.exp_sub]\n rw [integral_div, hγ, div_self (Real.exp_ne_zero γ)]\n have hYkint : ∀ k, Integrable (fun ω => Real.exp (r * X k ω - γ)) μ := by\n intro k; by_contra h\n have := hYint1 k\n rw [integral_undef h] at this\n exact zero_ne_one this\n rw [show (fun n ω => Real.exp (r * (∑ i ∈ Finset.range (n + 1), X i ω) - ((n : ℝ) + 1) * γ))\n = fun n ω => ∏ k ∈ Finset.range (n + 1), Real.exp (r * X k ω - γ) from\n by funext n; exact hZeq n]\n refine martingale_nat hAsm hint' (fun i => ?_)\n have hg : Integrable (fun ω => Real.exp (r * X (i + 1) ω - γ)) μ := hYkint (i + 1)\n have hfg : Integrable ((fun ω => ∏ k ∈ Finset.range (i + 1), Real.exp (r * X k ω - γ))\n * (fun ω => Real.exp (r * X (i + 1) ω - γ))) μ := by\n have he : ((fun ω => ∏ k ∈ Finset.range (i + 1), Real.exp (r * X k ω - γ))\n * (fun ω => Real.exp (r * X (i + 1) ω - γ)))\n = (fun ω => ∏ k ∈ Finset.range (i + 1 + 1), Real.exp (r * X k ω - γ)) := by\n funext ω; rw [Pi.mul_apply]\n exact (Finset.prod_range_succ (fun k => Real.exp (r * X k ω - γ)) (i + 1)).symm\n rw [he]; exact hint' (i + 1)\n have hcond : μ[fun ω => Real.exp (r * X (i + 1) ω - γ) | (ℱ i : MeasurableSpace Ω)]\n =ᵐ[μ] fun _ => ∫ ω, Real.exp (r * X (i + 1) ω - γ) ∂μ :=\n condExp_indep_eq (hmeas (i + 1)).measurable.comap_le (ℱ.le i)\n (hcont.comp_stronglyMeasurable (comap_measurable (X (i + 1))).stronglyMeasurable)\n (hindep.indep_comap_natural_of_lt hmeas (Nat.lt_succ_self i))\n have hstep : μ[fun ω => ∏ k ∈ Finset.range (i + 1 + 1), Real.exp (r * X k ω - γ)\n | (ℱ i : MeasurableSpace Ω)]\n =ᵐ[μ] fun ω => ∏ k ∈ Finset.range (i + 1), Real.exp (r * X k ω - γ) := by\n have he : (fun ω => ∏ k ∈ Finset.range (i + 1 + 1), Real.exp (r * X k ω - γ))\n = (fun ω => ∏ k ∈ Finset.range (i + 1), Real.exp (r * X k ω - γ))\n * (fun ω => Real.exp (r * X (i + 1) ω - γ)) := by\n funext ω; rw [Pi.mul_apply, Finset.prod_range_succ]\n rw [he]\n calc μ[(fun ω => ∏ k ∈ Finset.range (i + 1), Real.exp (r * X k ω - γ))\n * (fun ω => Real.exp (r * X (i + 1) ω - γ)) | (ℱ i : MeasurableSpace Ω)]\n =ᵐ[μ] (fun ω => ∏ k ∈ Finset.range (i + 1), Real.exp (r * X k ω - γ))\n * μ[fun ω => Real.exp (r * X (i + 1) ω - γ) | (ℱ i : MeasurableSpace Ω)] :=\n condExp_mul_of_aestronglyMeasurable_left (hAsm i).aestronglyMeasurable hfg hg\n _ =ᵐ[μ] fun ω => ∏ k ∈ Finset.range (i + 1), Real.exp (r * X k ω - γ) := by\n filter_upwards [hcond] with ω hω\n simp only [Pi.mul_apply, hω, hYint1 (i + 1), mul_one]\n exact hstep.symm", "time_seconds": 249, "topic": "Martingales & stopping" }, { "question_id": 267, "problem_name": "stochastic_processes_random_walk_martingale", "informal_statement": "Let (X_i) be i.i.d. with E[X_i] = 0, and let Z_n = X_1 + … + X_n.\n\nShow that (Z_n) is a martingale.", "lean_statement": "theorem q267_iid_sum_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hmeas : ∀ i, StronglyMeasurable (X i))\n (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (hint : ∀ i, Integrable (X i) μ) (hmean : ∫ ω, X 0 ω ∂μ = 0) :\n Martingale (fun n ω => ∑ i ∈ Finset.range (n + 1), X i ω)\n (Filtration.natural X hmeas) μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q267_iid_sum_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hmeas : ∀ i, StronglyMeasurable (X i))\n (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (hint : ∀ i, Integrable (X i) μ) (hmean : ∫ ω, X 0 ω ∂μ = 0) :\n Martingale (fun n ω => ∑ i ∈ Finset.range (n + 1), X i ω)\n (Filtration.natural X hmeas) μ := by\n set ℱ := Filtration.natural X hmeas with hℱ\n have hadapt : StronglyAdapted ℱ X := Filtration.stronglyAdapted_natural hmeas\n have hXmeas : ∀ i k, k ≤ i → StronglyMeasurable[ℱ i] (X k) := by\n intro i k hk\n exact (hadapt k).mono (ℱ.mono hk)\n have hXint0 : ∀ k, ∫ ω, X k ω ∂μ = 0 := fun k => (hident k).integral_eq.trans hmean\n have hsumfun : ∀ m : ℕ, (fun ω => ∑ k ∈ Finset.range (m + 1), X k ω)\n = ∑ k ∈ Finset.range (m + 1), X k := fun m => by funext ω; rw [Finset.sum_apply]\n refine ⟨?_, ?_⟩\n · intro n\n rw [show ((fun n ω => ∑ i ∈ Finset.range (n + 1), X i ω) n)\n = ∑ k ∈ Finset.range (n + 1), X k from hsumfun n]\n exact Finset.stronglyMeasurable_sum _\n (fun k hk => hXmeas n k (Nat.lt_succ_iff.mp (Finset.mem_range.mp hk)))\n · intro i j hij\n have hterm : ∀ k ∈ Finset.range (j + 1),\n (μ[X k | (ℱ i : MeasurableSpace Ω)]) =ᵐ[μ] fun ω => if k ≤ i then X k ω else 0 := by\n intro k _\n by_cases hki : k ≤ i\n · have heq : μ[X k | (ℱ i : MeasurableSpace Ω)] = X k :=\n condExp_of_stronglyMeasurable (ℱ.le i) (hXmeas i k hki) (hint k)\n filter_upwards with ω\n rw [if_pos hki]; exact congrFun heq ω\n · have hki' : i < k := not_le.mp hki\n have h0 : μ[X k | (ℱ i : MeasurableSpace Ω)] =ᵐ[μ] fun _ => ∫ ω, X k ω ∂μ :=\n hindep.condExp_natural_ae_eq_of_lt hmeas hki'\n filter_upwards [h0] with ω hω\n rw [hω, if_neg hki]\n exact hXint0 k\n have hall : ∀ᵐ ω ∂μ, ∀ k ∈ Finset.range (j + 1),\n (μ[X k | (ℱ i : MeasurableSpace Ω)]) ω = if k ≤ i then X k ω else 0 :=\n (eventually_all_finset _).2 hterm\n have hfilter : (Finset.range (j + 1)).filter (· ≤ i) = Finset.range (i + 1) := by\n ext k; simp only [Finset.mem_filter, Finset.mem_range]; omega\n have key : μ[(∑ k ∈ Finset.range (j + 1), X k) | (ℱ i : MeasurableSpace Ω)]\n =ᵐ[μ] (∑ k ∈ Finset.range (i + 1), X k) := by\n calc μ[(∑ k ∈ Finset.range (j + 1), X k) | (ℱ i : MeasurableSpace Ω)]\n =ᵐ[μ] ∑ k ∈ Finset.range (j + 1), μ[X k | (ℱ i : MeasurableSpace Ω)] :=\n condExp_finsetSum (fun k _ => hint k) (ℱ i : MeasurableSpace Ω)\n _ =ᵐ[μ] (∑ k ∈ Finset.range (i + 1), X k) := by\n filter_upwards [hall] with ω hω\n rw [Finset.sum_apply, Finset.sum_apply,\n Finset.sum_congr rfl (fun k hk => hω k hk), ← Finset.sum_filter, hfilter]\n show μ[fun ω => ∑ k ∈ Finset.range (j + 1), X k ω | (ℱ i : MeasurableSpace Ω)]\n =ᵐ[μ] fun ω => ∑ k ∈ Finset.range (i + 1), X k ω\n rw [hsumfun j, hsumfun i]\n exact key", "time_seconds": 579, "topic": "Martingales & stopping" }, { "question_id": 269, "problem_name": "stochastic_processes_slln_via_martingale", "informal_statement": "Let (X_i) be i.i.d. with mean X̄ and finite variance, and let S_n = X_1 + … + X_n.\n\nShow, using the Kolmogorov submartingale inequality and the martingale convergence theorem, that S_n/n → X̄ with probability 1.", "lean_statement": "theorem q269_slln_via_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (hL2 : MemLp (X 0) 2 μ)\n (xbar : ℝ) (hmean : ∫ ω, X 0 ω ∂μ = xbar)\n (S : ℕ → Ω → ℝ) (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, X i ω) :\n ∀ᵐ ω ∂μ, Tendsto (fun n => S n ω / (n : ℝ)) atTop (𝓝 xbar) := by\n sorry", "true_to_source": "literal", "proof": "theorem q269_slln_via_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (hL2 : MemLp (X 0) 2 μ)\n (xbar : ℝ) (hmean : ∫ ω, X 0 ω ∂μ = xbar)\n (S : ℕ → Ω → ℝ) (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, X i ω) :\n ∀ᵐ ω ∂μ, Tendsto (fun n => S n ω / (n : ℝ)) atTop (𝓝 xbar) := by\n have key := ProbabilityTheory.strong_law_ae_real X (hL2.integrable (by norm_num))\n (fun i j hij => hindep.indepFun hij) hident\n simp only [hmean] at key\n filter_upwards [key] with ω hω\n simpa only [hS] using hω", "time_seconds": 300, "topic": "Martingales & stopping" }, { "question_id": 272, "problem_name": "stochastic_processes_poisson_process_2", "informal_statement": "Customers enter a supermarket as a Poisson process with intensity 2.\n There are two salesmen near the door who offer passing customers samples of a new product.\n Each customer takes an exponential time of parameter 1 to think about the new product,\n and during this time occupies the full attention of one salesman.\n Having tried the product, customers proceed into the store and leave by another door.\n When both salesmen are occupied, customers walk straight in.\n \n Assuming that both salesmen are free at time 0,\n show that the probability that both are busy at a later time t is (4/15)e^{-5t}-(2/3)e^{-2t}+(2/5).", "lean_statement": "theorem q272_poisson_process_both_busy\n (p0 p1 p2 : ℝ → ℝ)\n (hd0 : Differentiable ℝ p0) (hd1 : Differentiable ℝ p1) (hd2 : Differentiable ℝ p2)\n (he0 : ∀ t, deriv p0 t = -2 * p0 t + p1 t)\n (he1 : ∀ t, deriv p1 t = 2 * p0 t - 3 * p1 t + 2 * p2 t)\n (he2 : ∀ t, deriv p2 t = 2 * p1 t - 2 * p2 t)\n (hi0 : p0 0 = 1) (hi1 : p1 0 = 0) (hi2 : p2 0 = 0)\n (t : ℝ) (ht : 0 ≤ t) :\n p2 t = (4 / 15) * Real.exp (-5 * t) - (2 / 3) * Real.exp (-2 * t) + 2 / 5 := by sorry", "true_to_source": "literal", "proof": "private noncomputable def A272 : (ℝ × ℝ × ℝ) →L[ℝ] (ℝ × ℝ × ℝ) :=\n let π0 : (ℝ × ℝ × ℝ) →L[ℝ] ℝ := ContinuousLinearMap.fst ℝ ℝ (ℝ × ℝ)\n let π1 : (ℝ × ℝ × ℝ) →L[ℝ] ℝ :=\n (ContinuousLinearMap.fst ℝ ℝ ℝ).comp (ContinuousLinearMap.snd ℝ ℝ (ℝ × ℝ))\n let π2 : (ℝ × ℝ × ℝ) →L[ℝ] ℝ :=\n (ContinuousLinearMap.snd ℝ ℝ ℝ).comp (ContinuousLinearMap.snd ℝ ℝ (ℝ × ℝ))\n (((-2 : ℝ) • π0 + π1).prod\n (((2:ℝ) • π0 - (3:ℝ) • π1 + (2:ℝ) • π2).prod ((2:ℝ) • π1 - (2:ℝ) • π2)))\n\nprivate lemma A272_apply (a b c : ℝ) :\n A272 (a, b, c) = (-2*a + b, 2*a - 3*b + 2*c, 2*b - 2*c) := by\n simp only [A272, ContinuousLinearMap.prod_apply, ContinuousLinearMap.add_apply,\n ContinuousLinearMap.sub_apply, ContinuousLinearMap.smul_apply,\n ContinuousLinearMap.coe_comp', Function.comp_apply, ContinuousLinearMap.coe_fst',\n ContinuousLinearMap.coe_snd', smul_eq_mul]\n\nprivate noncomputable def q0_272 (t : ℝ) : ℝ :=\n 1/5 + (2/3) * Real.exp (-2*t) + (2/15) * Real.exp (-5*t)\nprivate noncomputable def q1_272 (t : ℝ) : ℝ := 2/5 - (2/5) * Real.exp (-5*t)\nprivate noncomputable def q2_272 (t : ℝ) : ℝ :=\n 2/5 - (2/3) * Real.exp (-2*t) + (4/15) * Real.exp (-5*t)\n\nprivate lemma e2_272 (s : ℝ) :\n HasDerivAt (fun s : ℝ => Real.exp (-2*s)) (Real.exp (-2*s) * (-2)) s := by\n have h : HasDerivAt (fun s : ℝ => -2*s) (-2 : ℝ) s := by\n simpa using (hasDerivAt_id s).const_mul (-2 : ℝ)\n exact h.exp\n\nprivate lemma e5_272 (s : ℝ) :\n HasDerivAt (fun s : ℝ => Real.exp (-5*s)) (Real.exp (-5*s) * (-5)) s := by\n have h : HasDerivAt (fun s : ℝ => -5*s) (-5 : ℝ) s := by\n simpa using (hasDerivAt_id s).const_mul (-5 : ℝ)\n exact h.exp\n\nprivate lemma hq0_272 (s : ℝ) : HasDerivAt q0_272 (-2*q0_272 s + q1_272 s) s := by\n have hval : (-2*q0_272 s + q1_272 s)\n = (0 + (2/3) * (Real.exp (-2*s) * (-2)) + (2/15) * (Real.exp (-5*s) * (-5))) := by\n unfold q0_272 q1_272; ring\n rw [hval]\n exact ((hasDerivAt_const s (1/5:ℝ)).add ((e2_272 s).const_mul (2/3))).add\n ((e5_272 s).const_mul (2/15))\n\nprivate lemma hq1_272 (s : ℝ) : HasDerivAt q1_272 (2*q0_272 s - 3*q1_272 s + 2*q2_272 s) s := by\n have hval : (2*q0_272 s - 3*q1_272 s + 2*q2_272 s) = (0 - (2/5)*(Real.exp (-5*s)*(-5))) := by\n unfold q0_272 q1_272 q2_272; ring\n rw [hval]\n exact (hasDerivAt_const s (2/5:ℝ)).sub ((e5_272 s).const_mul (2/5))\n\nprivate lemma hq2_272 (s : ℝ) : HasDerivAt q2_272 (2*q1_272 s - 2*q2_272 s) s := by\n have hval : (2*q1_272 s - 2*q2_272 s)\n = ((0 - (2/3)*(Real.exp (-2*s)*(-2))) + (4/15)*(Real.exp (-5*s)*(-5))) := by\n unfold q1_272 q2_272; ring\n rw [hval]\n exact ((hasDerivAt_const s (2/5:ℝ)).sub ((e2_272 s).const_mul (2/3))).add\n ((e5_272 s).const_mul (4/15))\n\ntheorem q272_poisson_process_both_busy\n (p0 p1 p2 : ℝ → ℝ)\n (hd0 : Differentiable ℝ p0) (hd1 : Differentiable ℝ p1) (hd2 : Differentiable ℝ p2)\n (he0 : ∀ t, deriv p0 t = -2 * p0 t + p1 t)\n (he1 : ∀ t, deriv p1 t = 2 * p0 t - 3 * p1 t + 2 * p2 t)\n (he2 : ∀ t, deriv p2 t = 2 * p1 t - 2 * p2 t)\n (hi0 : p0 0 = 1) (hi1 : p1 0 = 0) (hi2 : p2 0 = 0)\n (t : ℝ) (ht : 0 ≤ t) :\n p2 t = (4 / 15) * Real.exp (-5 * t) - (2 / 3) * Real.exp (-2 * t) + 2 / 5 := by\n have hlip : ∀ _s : ℝ, LipschitzWith ‖A272‖₊ (fun x : ℝ×ℝ×ℝ => A272 x) := fun _ => A272.lipschitz\n have hfcont : ContinuousOn (fun s => (p0 s, p1 s, p2 s)) (Set.Icc (0:ℝ) t) :=\n (hd0.continuous.prodMk (hd1.continuous.prodMk hd2.continuous)).continuousOn\n have hgcont : ContinuousOn (fun s => (q0_272 s, q1_272 s, q2_272 s)) (Set.Icc (0:ℝ) t) := by\n apply Continuous.continuousOn\n unfold q0_272 q1_272 q2_272\n fun_prop\n have hfderiv : ∀ s ∈ Set.Ico (0:ℝ) t,\n HasDerivWithinAt (fun s => (p0 s, p1 s, p2 s)) (A272 (p0 s, p1 s, p2 s)) (Set.Ici s) s := by\n intro s _\n rw [A272_apply]\n have h0 : HasDerivAt p0 (-2*p0 s + p1 s) s := by rw [← he0 s]; exact (hd0 s).hasDerivAt\n have h1 : HasDerivAt p1 (2*p0 s - 3*p1 s + 2*p2 s) s := by\n rw [← he1 s]; exact (hd1 s).hasDerivAt\n have h2 : HasDerivAt p2 (2*p1 s - 2*p2 s) s := by rw [← he2 s]; exact (hd2 s).hasDerivAt\n exact (h0.prodMk (h1.prodMk h2)).hasDerivWithinAt\n have hgderiv : ∀ s ∈ Set.Ico (0:ℝ) t,\n HasDerivWithinAt (fun s => (q0_272 s, q1_272 s, q2_272 s))\n (A272 (q0_272 s, q1_272 s, q2_272 s)) (Set.Ici s) s := by\n intro s _\n rw [A272_apply]\n exact ((hq0_272 s).prodMk ((hq1_272 s).prodMk (hq2_272 s))).hasDerivWithinAt\n have ha : (fun s => (p0 s, p1 s, p2 s)) 0 = (fun s => (q0_272 s, q1_272 s, q2_272 s)) 0 := by\n simp only [hi0, hi1, hi2]\n unfold q0_272 q1_272 q2_272\n norm_num\n have hEq := ODE_solution_unique (v := fun _ x => A272 x) hlip hfcont hfderiv hgcont hgderiv ha\n have hfin := hEq (Set.mem_Icc.mpr ⟨ht, le_refl t⟩)\n simp only at hfin\n have h2eq : p2 t = q2_272 t := (Prod.ext_iff.mp (Prod.ext_iff.mp hfin).2).2\n rw [h2eq]\n unfold q2_272\n ring", "time_seconds": 460, "topic": "Continuous-time Markov & queues" }, { "question_id": 273, "problem_name": "stochastic_processes_ctmc_two_state_transition", "informal_statement": "Consider a two-state continuous-time Markov chain with transition rate λ from state 0 to state 1 and rate μ from state 1 to state 0.\n\nShow that P_{01}(t) = (λ/(λ+μ)) (1 − e^{−(λ+μ)t}).", "lean_statement": "theorem q273_ctmc_two_state_transition\n (lam mu : ℝ) (hlam : 0 < lam) (hmu : 0 < mu)\n (P01 : ℝ → ℝ) (hP0 : P01 0 = 0)\n (hODE : ∀ t : ℝ, HasDerivAt P01 (lam - (lam + mu) * P01 t) t) :\n ∀ t : ℝ, P01 t = (lam / (lam + mu)) * (1 - Real.exp (-(lam + mu) * t)) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q273_ctmc_two_state_transition\n (lam mu : ℝ) (hlam : 0 < lam) (hmu : 0 < mu)\n (P01 : ℝ → ℝ) (hP0 : P01 0 = 0)\n (hODE : ∀ t : ℝ, HasDerivAt P01 (lam - (lam + mu) * P01 t) t) :\n ∀ t : ℝ, P01 t = (lam / (lam + mu)) * (1 - Real.exp (-(lam + mu) * t)) := by\n set c := lam + mu with hc_def\n have hc : 0 < c := by rw [hc_def]; linarith\n have hcne : c ≠ 0 := ne_of_gt hc\n set g : ℝ → ℝ := fun t => (lam / c) * (1 - Real.exp (-c * t)) with hg_def\n have hg : ∀ t, HasDerivAt g (lam * Real.exp (-c * t)) t := by\n intro t\n have hmct : HasDerivAt (fun s => -c * s) (-c) t := by\n simpa using (hasDerivAt_id t).const_mul (-c)\n have hexp : HasDerivAt (fun s => Real.exp (-c * s)) (Real.exp (-c * t) * (-c)) t := hmct.exp\n have hsub : HasDerivAt (fun s => 1 - Real.exp (-c * s)) (0 - Real.exp (-c * t) * (-c)) t :=\n (hasDerivAt_const t (1 : ℝ)).sub hexp\n have hg0 : HasDerivAt g ((lam / c) * (0 - Real.exp (-c * t) * (-c))) t := by\n simpa [hg_def] using hsub.const_mul (lam / c)\n convert hg0 using 1\n field_simp\n ring\n have hh : ∀ t, HasDerivAt (fun s => Real.exp (c * s) * (P01 s - g s)) 0 t := by\n intro t\n have hct : HasDerivAt (fun s => c * s) c t := by\n simpa using (hasDerivAt_id t).const_mul c\n have hA : HasDerivAt (fun s => Real.exp (c * s)) (Real.exp (c * t) * c) t := hct.exp\n have hB : HasDerivAt (fun s => P01 s - g s)\n ((lam - c * P01 t) - lam * Real.exp (-c * t)) t := (hODE t).sub (hg t)\n have hmul := hA.mul hB\n convert hmul using 1\n simp only [hg_def]\n field_simp\n ring\n have hdiff : Differentiable ℝ (fun s => Real.exp (c * s) * (P01 s - g s)) :=\n fun t => (hh t).differentiableAt\n have hd0 : ∀ t, deriv (fun s => Real.exp (c * s) * (P01 s - g s)) t = 0 :=\n fun t => (hh t).deriv\n have hconst : ∀ t, (fun s => Real.exp (c * s) * (P01 s - g s)) t\n = (fun s => Real.exp (c * s) * (P01 s - g s)) 0 :=\n fun t => is_const_of_deriv_eq_zero hdiff hd0 t 0\n intro t\n have key := hconst t\n simp only at key\n have hg0val : g 0 = 0 := by simp [hg_def]\n rw [hP0, hg0val] at key\n simp only [mul_zero, sub_zero, Real.exp_zero, one_mul, sub_self, mul_zero] at key\n have hexp_pos : Real.exp (c * t) ≠ 0 := Real.exp_ne_zero _\n have hzero : P01 t - g t = 0 := by\n rcases mul_eq_zero.mp key with h | h\n · exact absurd h hexp_pos\n · exact h\n have : P01 t = g t := by linarith [hzero]\n exact this", "time_seconds": 226, "topic": "Continuous-time Markov & queues" }, { "question_id": 277, "problem_name": "stochastic_processes_optional_stopping_bounded_time", "informal_statement": "Let (X_n) be a supermartingale and let τ be a stopping time with τ ≤ M almost surely for some constant M.\n\nShow that E[X_τ] ≤ E[X_0].", "lean_statement": "theorem q277_optional_stopping_bounded_time (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ)\n (hX : Supermartingale X ℱ μ) (T : Ω → ℕ) (hT : IsStoppingTime ℱ (natStop T))\n (N : ℕ) (hTbdd : ∀ ω, T ω ≤ N) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ ≤ ∫ ω, X 0 ω ∂μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q277_optional_stopping_bounded_time (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ)\n (hX : Supermartingale X ℱ μ) (T : Ω → ℕ) (hT : IsStoppingTime ℱ (natStop T))\n (N : ℕ) (hTbdd : ∀ ω, T ω ≤ N) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ ≤ ∫ ω, X 0 ω ∂μ := by\n have hS0 : IsStoppingTime ℱ (natStop (fun _ : Ω => 0)) := by\n have : natStop (fun _ : Ω => (0:ℕ)) = fun _ : Ω => (0 : WithTop ℕ) := by\n funext ω; simp [natStop]\n rw [this]; exact isStoppingTime_const ℱ 0\n have hle : natStop (fun _ : Ω => 0) ≤ natStop T := fun ω => by\n unfold natStop; exact WithTop.coe_le_coe.mpr (Nat.zero_le _)\n have hbd : ∀ ω, natStop T ω ≤ (N : WithTop ℕ) := fun ω => by\n unfold natStop; exact WithTop.coe_le_coe.mpr (hTbdd ω)\n have key := (hX.neg).expected_stoppedValue_mono hS0 hT hle hbd\n have e1 : μ[stoppedValue (-X) (natStop (fun _ : Ω => 0))]\n = -μ[stoppedValue X (natStop (fun _ : Ω => 0))] := by\n simp only [stoppedValue, Pi.neg_apply, integral_neg]\n have e2 : μ[stoppedValue (-X) (natStop T)] = -μ[stoppedValue X (natStop T)] := by\n simp only [stoppedValue, Pi.neg_apply, integral_neg]\n rw [e1, e2] at key\n have e3 : stoppedValue X (natStop (fun _ : Ω => 0)) = X 0 := by\n funext ω; simp [stoppedValue, natStop]\n rw [e3] at key\n linarith", "time_seconds": 184, "topic": "Martingales & stopping" }, { "question_id": 278, "problem_name": "stochastic_processes_martingale_3", "informal_statement": "Let (ξ_i)_{i ≥ 1} be i.i.d with 𝔼[ξ1] = 1.\n\n Show that X_n = product_1^n ξ_i is a martingale.", "lean_statement": "theorem q278_martingale_product (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (ξ : ℕ → Ω → ℝ)\n (hint : ∀ n, Integrable (fun ω => ∏ i ∈ Finset.range n, ξ i ω) μ)\n (hmeas : ∀ n, StronglyMeasurable[ℱ (n + 1)] (ξ n))\n (hmean : ∀ n, μ[ξ n | ℱ n] =ᵐ[μ] 1) :\n Martingale (fun n ω => ∏ i ∈ Finset.range n, ξ i ω) ℱ μ := by sorry", "true_to_source": "literal", "proof": "theorem q278_martingale_product (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (ξ : ℕ → Ω → ℝ)\n (hint : ∀ n, Integrable (fun ω => ∏ i ∈ Finset.range n, ξ i ω) μ)\n (hmeas : ∀ n, StronglyMeasurable[ℱ (n + 1)] (ξ n))\n (hmean : ∀ n, μ[ξ n | ℱ n] =ᵐ[μ] 1) :\n Martingale (fun n ω => ∏ i ∈ Finset.range n, ξ i ω) ℱ μ := by\n -- adaptedness\n have hadapt : ∀ n, StronglyMeasurable[ℱ n] (fun ω => ∏ i ∈ Finset.range n, ξ i ω) := by\n intro n\n have heq : (fun ω => ∏ i ∈ Finset.range n, ξ i ω) = ∏ i ∈ Finset.range n, ξ i := by\n funext ω; rw [Finset.prod_apply]\n rw [heq]\n refine Finset.stronglyMeasurable_prod _ (fun i hi => ?_)\n have hle : ℱ (i + 1) ≤ ℱ n := ℱ.mono (by simp only [Finset.mem_range] at hi; omega)\n exact (hmeas i).mono hle\n -- integrability of each ξ n\n have hξint : ∀ n, Integrable (ξ n) μ := by\n intro n\n by_contra hcon\n have h0 : μ[ξ n | ℱ n] = 0 := condExp_of_not_integrable hcon\n have hae := hmean n\n rw [h0] at hae\n obtain ⟨ω, hω⟩ := hae.exists\n simp only [Pi.zero_apply, Pi.one_apply] at hω\n exact zero_ne_one hω\n -- one-step condition\n have hstep : ∀ i, (fun ω => ∏ j ∈ Finset.range i, ξ j ω) =ᵐ[μ]\n μ[(fun ω => ∏ j ∈ Finset.range (i + 1), ξ j ω) | ℱ i] := by\n intro i\n have hXsucc : (fun ω => ∏ j ∈ Finset.range (i + 1), ξ j ω)\n = (fun ω => ∏ j ∈ Finset.range i, ξ j ω) * ξ i := by\n funext ω; rw [Finset.prod_range_succ, Pi.mul_apply]\n have hf : StronglyMeasurable[ℱ i] (fun ω => ∏ j ∈ Finset.range i, ξ j ω) := hadapt i\n have hfg : Integrable ((fun ω => ∏ j ∈ Finset.range i, ξ j ω) * ξ i) μ := by\n rw [← hXsucc]; exact hint (i + 1)\n have hpull := condExp_mul_of_stronglyMeasurable_left hf hfg (hξint i)\n have key : μ[(fun ω => ∏ j ∈ Finset.range (i + 1), ξ j ω) | ℱ i]\n =ᵐ[μ] (fun ω => ∏ j ∈ Finset.range i, ξ j ω) := by\n rw [hXsucc]\n refine hpull.trans ?_\n filter_upwards [hmean i] with ω hω\n have hω1 : μ[ξ i | ℱ i] ω = 1 := by simpa using hω\n simp only [Pi.mul_apply, hω1, mul_one]\n exact key.symm\n exact martingale_nat hadapt hint hstep", "time_seconds": 232, "topic": "Martingales & stopping" }, { "question_id": 281, "problem_name": "stochastic_processes_renewal_count_strictly_less", "informal_statement": "Let (N(t)) be a renewal process with arrival epochs S_n.\n\nShow that {N(t) ≥ n} = {S_n ≤ t}, so that N(t) = max{n : S_n ≤ t}.", "lean_statement": "theorem q281_renewal_count_strictly_less (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hpos : ∀ i ω, 0 < X i ω)\n (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (S : ℕ → Ω → ℝ) (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, X i ω)\n (N : ℝ → Ω → ℕ)\n (hgreatest : ∀ (t : ℝ) (ω : Ω), 0 ≤ t → IsGreatest {n : ℕ | S n ω ≤ t} (N t ω)) :\n ∀ (n : ℕ) (t : ℝ), 0 ≤ t → {ω | n ≤ N t ω} = {ω | S n ω ≤ t} := by\n sorry", "true_to_source": "abstract", "proof": "theorem q281_renewal_count_strictly_less (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hpos : ∀ i ω, 0 < X i ω)\n (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (S : ℕ → Ω → ℝ) (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, X i ω)\n (N : ℝ → Ω → ℕ)\n (hgreatest : ∀ (t : ℝ) (ω : Ω), 0 ≤ t → IsGreatest {n : ℕ | S n ω ≤ t} (N t ω)) :\n ∀ (n : ℕ) (t : ℝ), 0 ≤ t → {ω | n ≤ N t ω} = {ω | S n ω ≤ t} := by\n intro n t ht\n ext ω\n simp only [Set.mem_setOf_eq]\n have hmono : ∀ a b : ℕ, a ≤ b → S a ω ≤ S b ω := by\n intro a b hab\n rw [hS, hS]\n apply Finset.sum_le_sum_of_subset_of_nonneg\n · exact fun x hx => Finset.mem_range.mpr (lt_of_lt_of_le (Finset.mem_range.mp hx) hab)\n · intro i _ _\n exact le_of_lt (hpos i ω)\n constructor\n · intro hn\n have h1 : S (N t ω) ω ≤ t := (hgreatest t ω ht).1\n exact le_trans (hmono n (N t ω) hn) h1\n · intro hn\n exact (hgreatest t ω ht).2 hn", "time_seconds": 131, "topic": "Renewal processes" }, { "question_id": 283, "problem_name": "stochastic_processes_gamblers_ruin_3", "informal_statement": "Let X = (X_n)_{n ≥ 0} be Gambler’s ruin with state space Ω = {0, 1, 2, ..., N}:\n X_0 = k, ℙ[X_{n + 1} = X_n + 1 | X_n] = 1/2, τ = min{n: X_n = 0 or N}.\n\n Show that 𝔼[τ] = k(N - k).", "lean_statement": "theorem q283_gambler_expected_time (N : ℕ) (hN : 0 < N) (e : ℕ → ℝ)\n (hb0 : e 0 = 0) (hbN : e N = 0)\n (hrec : ∀ k, 0 < k → k < N → e k = 1 + (e (k - 1) + e (k + 1)) / 2)\n (k : ℕ) (hk : k ≤ N) :\n e k = (k : ℝ) * ((N : ℝ) - (k : ℝ)) := by sorry", "true_to_source": "literal", "proof": "theorem q283_gambler_expected_time (N : ℕ) (hN : 0 < N) (e : ℕ → ℝ)\n (hb0 : e 0 = 0) (hbN : e N = 0)\n (hrec : ∀ k, 0 < k → k < N → e k = 1 + (e (k - 1) + e (k + 1)) / 2)\n (k : ℕ) (hk : k ≤ N) :\n e k = (k : ℝ) * ((N : ℝ) - (k : ℝ)) := by\n set d1 : ℝ := e 1 - ((N:ℝ) - 1) with hd1\n have key : ∀ j, j ≤ N → e j = (j:ℝ)*((N:ℝ)-(j:ℝ)) + (j:ℝ)*d1 := by\n intro j\n induction j using Nat.strong_induction_on with\n | _ j IH =>\n rcases j with _ | _ | m\n · intro _; simp [hb0]\n · intro _; push_cast; rw [hd1]; ring\n · intro hjN\n simp only [Nat.succ_eq_add_one] at *\n have hr := hrec (m+1) (by omega) (by omega)\n simp only [Nat.add_sub_cancel] at hr\n have e1 : e (m+1) = ((m+1:ℕ):ℝ)*((N:ℝ)-((m+1:ℕ):ℝ)) + ((m+1:ℕ):ℝ)*d1 :=\n IH (m+1) (by omega) (by omega)\n have e0 : e m = ((m:ℕ):ℝ)*((N:ℝ)-((m:ℕ):ℝ)) + ((m:ℕ):ℝ)*d1 :=\n IH m (by omega) (by omega)\n push_cast at e1 e0 ⊢\n linear_combination (-2 : ℝ) * hr + 2 * e1 - e0\n have hkey := key k hk\n have hN0 := key N le_rfl\n rw [hbN] at hN0\n have hNd : (N:ℝ) * d1 = 0 := by\n have h0 : (N:ℝ) * ((N:ℝ) - (N:ℝ)) = 0 := by ring\n linarith [hN0]\n have hNne : (N:ℝ) ≠ 0 := by\n have : N ≠ 0 := by omega\n exact_mod_cast this\n have hd10 : d1 = 0 := by\n rcases mul_eq_zero.mp hNd with h | h\n · exact absurd h hNne\n · exact h\n rw [hkey, hd10]; ring", "time_seconds": 227, "topic": "Random walks & large deviations" }, { "question_id": 285, "problem_name": "stochastic_processes_renewal_borel_cantelli_lemma", "informal_statement": "Let (Y_n) be a sequence of random variables with Σ_{n≥1} E[|Y_n|] < ∞.\n\nShow that Pr[lim_{n→∞} Y_n = 0] = 1.", "lean_statement": "theorem q285_borel_cantelli (μ : Measure Ω) [IsProbabilityMeasure μ]\n (Y : ℕ → Ω → ℝ) (hY : ∀ n, AEMeasurable (Y n) μ)\n (hYint : ∀ n, Integrable (Y n) μ)\n (hsum : Summable (fun n => ∫ ω, |Y n ω| ∂μ)) :\n ∀ᵐ ω ∂μ, Tendsto (fun n => Y n ω) atTop (𝓝 0) := by\n sorry", "true_to_source": "literal", "proof": "theorem q285_borel_cantelli (μ : Measure Ω) [IsProbabilityMeasure μ]\n (Y : ℕ → Ω → ℝ) (hY : ∀ n, AEMeasurable (Y n) μ)\n (hYint : ∀ n, Integrable (Y n) μ)\n (hsum : Summable (fun n => ∫ ω, |Y n ω| ∂μ)) :\n ∀ᵐ ω ∂μ, Tendsto (fun n => Y n ω) atTop (𝓝 0) := by\n have hf'' : ∀ i, AEMeasurable (fun ω => ‖Y i ω‖ₑ) μ := fun i => (hY i).enorm\n have hlint : ∀ i, ∫⁻ ω, ‖Y i ω‖ₑ ∂μ = ENNReal.ofReal (∫ ω, |Y i ω| ∂μ) := by\n intro i\n rw [← ofReal_integral_norm_eq_lintegral_enorm (hYint i)]\n simp only [Real.norm_eq_abs]\n have hsum' : ∑' i, ∫⁻ ω, ‖Y i ω‖ₑ ∂μ ≠ ∞ := by\n simp_rw [hlint]\n rw [← ENNReal.ofReal_tsum_of_nonneg (fun i => integral_nonneg (fun ω => abs_nonneg _)) hsum]\n exact ENNReal.ofReal_ne_top\n have hae : ∀ᵐ ω ∂μ, Summable (fun n => ((‖Y n ω‖₊ : ℝ))) := by\n have hfin : ∫⁻ ω, ∑' n, ‖Y n ω‖ₑ ∂μ ≠ ∞ := by\n rw [lintegral_tsum hf'']; exact hsum'\n refine (ae_lt_top' (AEMeasurable.tsum hf'') hfin).mono ?_\n intro ω hω\n rw [← ENNReal.tsum_coe_ne_top_iff_summable_coe]\n simp_rw [← enorm_eq_nnnorm]\n exact hω.ne\n filter_upwards [hae] with ω hω\n rw [tendsto_zero_iff_norm_tendsto_zero]\n simpa [coe_nnnorm] using hω.tendsto_atTop_zero", "time_seconds": 190, "topic": "Weak convergence & functional limits" }, { "question_id": 291, "problem_name": "stochastic_processes_martingale_4", "informal_statement": "Let X = (X_n)_{n ≥ 0} be a martingale in L^2.\n\n Show that its increments (X_{n + 1} − X_n)_{n ≥ 0} are pairwise orthogonal,\n i.e. for all n ≠ m, we have 𝔼[(X_{n + 1} − X_n)(X_{m + 1} − X_m)] = 0.", "lean_statement": "theorem q291_l2_orthogonal (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hX : Martingale X ℱ μ) (hL2 : ∀ n, MemLp (X n) 2 μ) (n m : ℕ) (hnm : n ≠ m) :\n ∫ ω, (X (n + 1) ω - X n ω) * (X (m + 1) ω - X m ω) ∂μ = 0 := by sorry", "true_to_source": "literal", "proof": "theorem q291_l2_orthogonal (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hX : Martingale X ℱ μ) (hL2 : ∀ n, MemLp (X n) 2 μ) (n m : ℕ) (hnm : n ≠ m) :\n ∫ ω, (X (n + 1) ω - X n ω) * (X (m + 1) ω - X m ω) ∂μ = 0 := by\n have core : ∀ a b : ℕ, a < b →\n ∫ ω, (X (b + 1) ω - X b ω) * (X (a + 1) ω - X a ω) ∂μ = 0 := by\n intro a b hab\n -- f = X(a+1) - X a is ℱ b measurable\n have hf_meas : StronglyMeasurable[ℱ b] (fun ω => X (a+1) ω - X a ω) :=\n StronglyMeasurable.sub ((hX.stronglyAdapted (a+1)).mono (ℱ.mono (by omega)))\n ((hX.stronglyAdapted a).mono (ℱ.mono (by omega)))\n have hfL2 : MemLp (fun ω => X (a+1) ω - X a ω) 2 μ := (hL2 (a+1)).sub (hL2 a)\n have hgL2 : MemLp (fun ω => X (b+1) ω - X b ω) 2 μ := (hL2 (b+1)).sub (hL2 b)\n have hf_int : Integrable (fun ω => X (a+1) ω - X a ω) μ := hfL2.integrable (by norm_num)\n have hg_int : Integrable (fun ω => X (b+1) ω - X b ω) μ := hgL2.integrable (by norm_num)\n have hfg_int : Integrable\n ((fun ω => X (a+1) ω - X a ω) * (fun ω => X (b+1) ω - X b ω)) μ :=\n hfL2.integrable_mul hgL2\n -- condExp of g = X(b+1)-X b at ℱ b is 0\n have hmg : μ[X (b+1) | ℱ b] =ᵐ[μ] X b := hX.condExp_ae_eq (by omega)\n have hmb : μ[X b | ℱ b] = X b :=\n condExp_of_stronglyMeasurable (ℱ.le b) (hX.stronglyAdapted b) ((hL2 b).integrable (by norm_num))\n have hg_ce : μ[fun ω => X (b+1) ω - X b ω | ℱ b] =ᵐ[μ] 0 := by\n have hsub : μ[(X (b+1)) - (X b) | ℱ b] =ᵐ[μ] μ[X (b+1)|ℱ b] - μ[X b|ℱ b] :=\n condExp_sub ((hL2 (b+1)).integrable (by norm_num)) ((hL2 b).integrable (by norm_num)) _\n filter_upwards [hsub, hmg] with ω h1 h2\n simp only [Pi.sub_apply, hmb] at h1 ⊢\n rw [show (fun ω => X (b+1) ω - X b ω) = (X (b+1)) - (X b) from rfl, h1, h2]\n simp\n -- pull out the ℱ b-measurable factor\n have hpull : μ[(fun ω => X (a+1) ω - X a ω) * (fun ω => X (b+1) ω - X b ω) | ℱ b]\n =ᵐ[μ] 0 := by\n refine (condExp_mul_of_stronglyMeasurable_left hf_meas hfg_int hg_int).trans ?_\n filter_upwards [hg_ce] with ω hω\n simp only [Pi.mul_apply, hω, Pi.zero_apply, mul_zero]\n -- integrate\n calc ∫ ω, (X (b+1) ω - X b ω) * (X (a+1) ω - X a ω) ∂μ\n = ∫ ω, (X (a+1) ω - X a ω) * (X (b+1) ω - X b ω) ∂μ := by\n apply integral_congr_ae; filter_upwards with ω; ring\n _ = ∫ ω, μ[(fun ω => X (a+1) ω - X a ω) * (fun ω => X (b+1) ω - X b ω) | ℱ b] ω ∂μ :=\n (integral_condExp (ℱ.le b)).symm\n _ = ∫ ω, (0:ℝ) ∂μ := integral_congr_ae hpull\n _ = 0 := by simp\n rcases lt_or_gt_of_ne hnm with h | h\n · rw [← core n m h]\n apply integral_congr_ae; filter_upwards with ω; ring\n · exact core m n h", "time_seconds": 247, "topic": "Martingales & stopping" }, { "question_id": 295, "problem_name": "stochastic_processes_mdp_aggregate_reward_recursion", "informal_statement": "Let (X_n) be a finite-state Markov chain with transition matrix [P] and a reward r_i earned on each visit to state i. Let v_i(n) be the expected aggregate reward over n transitions starting in state i.\n\nShow that the vector v(n) satisfies the recursion v(n) = r + [P] v(n−1) with v(0) = 0.", "lean_statement": "theorem q295_aggregate_reward_recursion {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (r : Ω → ℝ)\n (v : ℕ → Ω → ℝ)\n (hv : ∀ n i, v n i = ∑ m ∈ Finset.range n, ∑ j, (P ^ m) i j * r j) :\n (∀ i, v 0 i = 0) ∧ (∀ n i, v (n + 1) i = r i + ∑ j, P i j * v n j) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q295_aggregate_reward_recursion {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (r : Ω → ℝ)\n (v : ℕ → Ω → ℝ)\n (hv : ∀ n i, v n i = ∑ m ∈ Finset.range n, ∑ j, (P ^ m) i j * r j) :\n (∀ i, v 0 i = 0) ∧ (∀ n i, v (n + 1) i = r i + ∑ j, P i j * v n j) := by\n constructor\n · intro i\n rw [hv]\n simp\n · intro n i\n rw [hv (n + 1) i]\n have key : ∑ j, P i j * v n j = ∑ m ∈ Finset.range n, ∑ j, (P ^ (m + 1)) i j * r j := by\n simp_rw [hv n, Finset.mul_sum]\n rw [Finset.sum_comm]\n apply Finset.sum_congr rfl\n intro m _\n rw [Finset.sum_comm]\n apply Finset.sum_congr rfl\n intro k _\n have : (∑ j, P i j * ((P ^ m) j k * r k)) = (∑ j, P i j * (P ^ m) j k) * r k := by\n rw [Finset.sum_mul]\n apply Finset.sum_congr rfl\n intro j _\n ring\n rw [this, pow_succ', Matrix.mul_apply]\n rw [key, Finset.sum_range_succ']\n have hzero : (∑ j, (P ^ 0) i j * r j) = r i := by\n simp [Matrix.one_apply, Finset.sum_ite_eq]\n rw [hzero, add_comm]", "time_seconds": 170, "topic": "Markov chains (finite & countable)" }, { "question_id": 297, "problem_name": "stochastic_processes_rate_function_normal", "informal_statement": "Let X be a standard normal random variable and let I(a)=sup_θ(θa − log M(θ)).\n\nShow that I(a) = a^2/2.", "lean_statement": "theorem q297_rate_function_normal (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : Ω → ℝ) (hlaw : HasLaw X (gaussianReal 0 1) μ) (a : ℝ) :\n sSup ((fun θ : ℝ => θ * a - cgf X μ θ) ''\n {θ : ℝ | Integrable (fun ω => Real.exp (θ * X ω)) μ})\n = a ^ 2 / 2 := by\n sorry", "true_to_source": "literal", "proof": "theorem q297_rate_function_normal (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : Ω → ℝ) (hlaw : HasLaw X (gaussianReal 0 1) μ) (a : ℝ) :\n sSup ((fun θ : ℝ => θ * a - cgf X μ θ) ''\n {θ : ℝ | Integrable (fun ω => Real.exp (θ * X ω)) μ})\n = a ^ 2 / 2 := by\n have hmap := hlaw.map_eq\n have hcgf : ∀ θ : ℝ, cgf X μ θ = θ ^ 2 / 2 := by\n intro θ\n have := cgf_gaussianReal hmap θ\n simpa using this\n have hint : ∀ t : ℝ, Integrable (fun ω => Real.exp (t * X ω)) μ := by\n intro t\n have h1 : Integrable (fun x => Real.exp (t * x)) (μ.map X) := by\n rw [hmap]; exact integrable_exp_mul_gaussianReal t\n exact (integrable_map_measure (by fun_prop) hlaw.aemeasurable).mp h1\n have hgreat : IsGreatest ((fun θ : ℝ => θ * a - cgf X μ θ) ''\n {θ : ℝ | Integrable (fun ω => Real.exp (θ * X ω)) μ}) (a ^ 2 / 2) := by\n constructor\n · refine ⟨a, hint a, ?_⟩\n simp only [hcgf]; ring\n · rintro y ⟨θ, hθ, rfl⟩\n simp only [hcgf]\n nlinarith [sq_nonneg (a - θ)]\n exact hgreat.csSup_eq", "time_seconds": 185, "topic": "Random walks & large deviations" }, { "question_id": 301, "problem_name": "stochastic_processes_supermartingale_stopping_time_2", "informal_statement": "Let (X_n)_{n ≥ 0} be a supermartingale, and let T be a stopping time.\n\n Show that X^T is also a supermartingale.", "lean_statement": "theorem q301_stopped_supermartingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Supermartingale X ℱ μ) (T : Ω → ℕ)\n (hT : IsStoppingTime ℱ (natStop T)) :\n Supermartingale (stoppedProcess X (natStop T)) ℱ μ := by sorry", "true_to_source": "literal", "proof": "theorem q301_stopped_supermartingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Supermartingale X ℱ μ) (T : Ω → ℕ)\n (hT : IsStoppingTime ℱ (natStop T)) :\n Supermartingale (stoppedProcess X (natStop T)) ℱ μ := by\n have hsub : Submartingale (stoppedProcess (-X) (natStop T)) ℱ μ :=\n hX.neg.stoppedProcess hT\n have hkey : -(stoppedProcess (-X) (natStop T)) = stoppedProcess X (natStop T) := by\n funext i ω\n simp [stoppedProcess, Pi.neg_apply]\n have := hsub.neg\n rwa [hkey] at this", "time_seconds": 42, "topic": "Martingales & stopping" }, { "question_id": 302, "problem_name": "stochastic_processes_kolmogorov_inequality_generalized", "informal_statement": "Let (Z_n) be a nonnegative submartingale and let a > 0.\n\nShow that a Pr[max_{1 ≤ i ≤ m} Z_i ≥ a] ≤ E[Z_m 1{max_{1 ≤ i ≤ m} Z_i ≥ a}], and deduce the Kolmogorov submartingale inequality from it.", "lean_statement": "theorem q302_kolmogorov_inequality_generalized (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (Z : ℕ → Ω → ℝ) (hZ : Submartingale Z ℱ μ)\n (hnn : ∀ n ω, 0 ≤ Z n ω) (a : ℝ) (ha : 0 < a) (m : ℕ) (hm : 1 ≤ m) :\n a * (μ {ω | a ≤ runningMax Z m ω}).toReal\n ≤ ∫ ω in {ω | a ≤ runningMax Z m ω}, Z m ω ∂μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q302_kolmogorov_inequality_generalized (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (Z : ℕ → Ω → ℝ) (hZ : Submartingale Z ℱ μ)\n (hnn : ∀ n ω, 0 ≤ Z n ω) (a : ℝ) (ha : 0 < a) (m : ℕ) (hm : 1 ≤ m) :\n a * (μ {ω | a ≤ runningMax Z m ω}).toReal\n ≤ ∫ ω in {ω | a ≤ runningMax Z m ω}, Z m ω ∂μ := by\n have hrm : ∀ ω, runningMax Z m ω\n = (Finset.range (m+1)).sup' (by simp) (fun k => Z k ω) := by\n intro ω\n unfold runningMax\n rw [← Finset.sup'_univ_eq_ciSup]\n apply le_antisymm\n · apply Finset.sup'_le; intro k _\n apply Finset.le_sup' (fun k => Z k ω); simp [Finset.mem_range, k.isLt]\n · apply Finset.sup'_le; intro j hj\n rw [Finset.mem_range] at hj\n exact Finset.le_sup' (fun k : Fin (m+1) => Z (↑k) ω) (Finset.mem_univ ⟨j, hj⟩)\n have hcoe : ((a.toNNReal : ℝ≥0) : ℝ) = a := Real.coe_toNNReal a ha.le\n have hnn' : (0 : ℕ → Ω → ℝ) ≤ Z := fun n ω => hnn n ω\n have key := MeasureTheory.maximal_ineq hZ hnn' (ε := a.toNNReal) m\n rw [hcoe] at key\n simp only [← hrm] at key\n -- key : a.toNNReal • μ S ≤ ENNReal.ofReal (∫ ω in S, Z m ω ∂μ)\n have hI : 0 ≤ ∫ ω in {ω | a ≤ runningMax Z m ω}, Z m ω ∂μ :=\n integral_nonneg (fun ω => hnn m ω)\n have key2 := ENNReal.toReal_mono ENNReal.ofReal_ne_top key\n rw [ENNReal.toReal_ofReal hI, ENNReal.toReal_mul, ENNReal.coe_toReal, hcoe] at key2\n exact key2", "time_seconds": 933, "topic": "Martingales & stopping" }, { "question_id": 304, "problem_name": "stochastic_processes_poisson_superposition_1", "informal_statement": "If X and Y are independent Poisson random variables with means a and b (the arrival counts of two independent Poisson processes), then X + Y is Poisson with mean a + b.", "lean_statement": "theorem q304_poisson_superposition (μ : Measure Ω) [IsProbabilityMeasure μ]\n (a b : ℝ≥0) (X Y : Ω → ℕ)\n (hX : HasLaw X (poissonMeasure a) μ) (hY : HasLaw Y (poissonMeasure b) μ)\n (hindep : IndepFun X Y μ) :\n HasLaw (fun ω => X ω + Y ω) (poissonMeasure (a + b)) μ := by\n sorry", "true_to_source": "abstract", "proof": "theorem q304_poisson_superposition (μ : Measure Ω) [IsProbabilityMeasure μ]\n (a b : ℝ≥0) (X Y : Ω → ℕ) (hX : HasLaw X (poissonMeasure a) μ)\n (hY : HasLaw Y (poissonMeasure b) μ) (hindep : IndepFun X Y μ) :\n HasLaw (fun ω => X ω + Y ω) (poissonMeasure (a + b)) μ :=\n hindep.hasLaw_add_poissonMeasure hX hY", "time_seconds": 86, "topic": "Poisson processes" }, { "question_id": 305, "problem_name": "stochastic_processes_martingale_transform_supermartingale", "informal_statement": "Let (X_n) be a supermartingale and let (H_n) be a non-negative predictable sequence.\n\nShow that Z_n = Σ_{m=1}^n H_m (X_m − X_{m−1}) is a supermartingale.", "lean_statement": "theorem q305_martingale_transform_supermartingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (X : ℕ → Ω → ℝ) (hX : Supermartingale X ℱ μ)\n (H : ℕ → Ω → ℝ) (R : ℝ)\n (hpred : StronglyAdapted ℱ (fun n => H (n + 1)))\n (hbdd : ∀ n ω, H n ω ≤ R) (hnonneg : ∀ n ω, 0 ≤ H n ω) :\n Supermartingale\n (fun n ω => ∑ m ∈ Finset.range n, H (m + 1) ω * (X (m + 1) ω - X m ω)) ℱ μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q305_martingale_transform_supermartingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (X : ℕ → Ω → ℝ) (hX : Supermartingale X ℱ μ)\n (H : ℕ → Ω → ℝ) (R : ℝ)\n (hpred : StronglyAdapted ℱ (fun n => H (n + 1)))\n (hbdd : ∀ n ω, H n ω ≤ R) (hnonneg : ∀ n ω, 0 ≤ H n ω) :\n Supermartingale\n (fun n ω => ∑ m ∈ Finset.range n, H (m + 1) ω * (X (m + 1) ω - X m ω)) ℱ μ := by\n have key := (hX.neg.sum_mul_sub' hpred hbdd hnonneg).neg\n have heq : (-fun n => ∑ k ∈ Finset.range n, H (k + 1) * ((-X) (k + 1) - (-X) k))\n = (fun n ω => ∑ m ∈ Finset.range n, H (m + 1) ω * (X (m + 1) ω - X m ω)) := by\n funext n ω\n simp only [Pi.neg_apply, Finset.sum_apply, Pi.mul_apply, Pi.sub_apply]\n rw [← Finset.sum_neg_distrib]\n exact Finset.sum_congr rfl fun k _ => by ring\n rw [heq] at key\n exact key", "time_seconds": 111, "topic": "Martingales & stopping" }, { "question_id": 311, "problem_name": "stochastic_processes_poisson_counting_marginal", "informal_statement": "Consider a Poisson process of rate r > 0: the increments over disjoint intervals satisfy N(b) − N(a) ~ Poisson(r(b − a)), and N(0) = 0. Show that for every t, the number of arrivals N(t) has a Poisson distribution with parameter r t.", "lean_statement": "theorem q311_poisson_marginal (μ : Measure Ω) [IsProbabilityMeasure μ]\n (r : ℝ≥0) (hr : 0 < r) (N : ℝ≥0 → Ω → ℕ)\n (hincr : ∀ a b : ℝ≥0, a ≤ b →\n HasLaw (fun ω => N b ω - N a ω) (poissonMeasure (r * (b - a))) μ)\n (hzero : ∀ᵐ ω ∂μ, N 0 ω = 0) :\n ∀ t : ℝ≥0, HasLaw (N t) (poissonMeasure (r * t)) μ := by\n sorry", "true_to_source": "abstract", "proof": "theorem q311_poisson_marginal (μ : Measure Ω) [IsProbabilityMeasure μ]\n (r : ℝ≥0) (hr : 0 < r) (N : ℝ≥0 → Ω → ℕ)\n (hincr : ∀ a b : ℝ≥0, a ≤ b →\n HasLaw (fun ω => N b ω - N a ω) (poissonMeasure (r * (b - a))) μ)\n (hzero : ∀ᵐ ω ∂μ, N 0 ω = 0) :\n ∀ t : ℝ≥0, HasLaw (N t) (poissonMeasure (r * t)) μ := by\n intro t\n have hlaw := hincr 0 t (by positivity)\n rw [tsub_zero] at hlaw\n refine hlaw.congr ?_\n filter_upwards [hzero] with ω h\n rw [h, Nat.sub_zero]", "time_seconds": 131, "topic": "Poisson processes" }, { "question_id": 318, "problem_name": "stochastic_processes_jackson_traffic_equations", "informal_statement": "Consider a Jackson network of k queues, where exogenous arrivals enter queue j as a Poisson process of rate λ_0 Q_{0j}, a customer completing service at queue i moves to queue j with probability Q_{ij} and departs the network with probability Q_{i0}.\n\nShow that the effective arrival rates λ_j satisfy the traffic equations λ_j = λ_0 Q_{0j} + Σ_{i=1}^k λ_i Q_{ij}, and that these have a unique solution when the network is open and every queue is reachable.", "lean_statement": "theorem q318_jackson_traffic_equations {k : ℕ}\n (lam0 : ℝ) (Q0 : Fin k → ℝ)\n (R : Matrix (Fin k) (Fin k) ℝ) (hR0 : ∀ i j, 0 ≤ R i j)\n (a : Fin k → ℝ) (ha : ∀ j, a j = lam0 * Q0 j)\n (hopen : IsUnit ((1 : Matrix (Fin k) (Fin k) ℝ) - Rᵀ)) :\n ∃! lam : Fin k → ℝ, ∀ j, lam j = lam0 * Q0 j + ∑ i, lam i * R i j := by\n sorry", "true_to_source": "abstract", "proof": "theorem q318_jackson_traffic_equations {k : ℕ}\n (lam0 : ℝ) (Q0 : Fin k → ℝ)\n (R : Matrix (Fin k) (Fin k) ℝ) (hR0 : ∀ i j, 0 ≤ R i j)\n (a : Fin k → ℝ) (ha : ∀ j, a j = lam0 * Q0 j)\n (hopen : IsUnit ((1 : Matrix (Fin k) (Fin k) ℝ) - Rᵀ)) :\n ∃! lam : Fin k → ℝ, ∀ j, lam j = lam0 * Q0 j + ∑ i, lam i * R i j := by\n set M : Matrix (Fin k) (Fin k) ℝ := (1 : Matrix (Fin k) (Fin k) ℝ) - Rᵀ with hM\n have hdet : IsUnit M.det := (Matrix.isUnit_iff_isUnit_det M).mp hopen\n have hRT : ∀ (lam : Fin k → ℝ) (j), (Rᵀ *ᵥ lam) j = ∑ i, lam i * R i j := by\n intro lam j\n rw [Matrix.mulVec_eq_sum]\n simp only [Finset.sum_apply, Pi.smul_apply, Matrix.transpose_apply]\n exact Finset.sum_congr rfl (fun i _ => mul_comm _ _)\n have key : ∀ lam : Fin k → ℝ,\n (∀ j, lam j = lam0 * Q0 j + ∑ i, lam i * R i j) ↔ M *ᵥ lam = fun j => lam0 * Q0 j := by\n intro lam\n rw [funext_iff]\n apply forall_congr'\n intro j\n have hcomp : (M *ᵥ lam) j = lam j - ∑ i, lam i * R i j := by\n rw [hM, Matrix.sub_mulVec, Pi.sub_apply, Matrix.one_mulVec, hRT]\n rw [hcomp]\n constructor <;> intro h <;> linarith [h]\n refine ⟨M⁻¹ *ᵥ (fun j => lam0 * Q0 j), ?_, ?_⟩\n · refine (key _).mpr ?_\n rw [Matrix.mulVec_mulVec, Matrix.mul_nonsing_inv M hdet, Matrix.one_mulVec]\n · intro y hy\n rw [key] at hy\n have h2 := congrArg (fun v => M⁻¹ *ᵥ v) hy\n simp only [Matrix.mulVec_mulVec, Matrix.nonsing_inv_mul M hdet, Matrix.one_mulVec] at h2\n exact h2", "time_seconds": 109, "topic": "Continuous-time Markov & queues" }, { "question_id": 321, "problem_name": "stochastic_processes_chain_coupling", "informal_statement": "Suppose that P is irreducible with stationary distribution π.\n Let (X_n, Y_n)_{n ≥ 0} be a coupling of Markov chains with\n transition matrix P for which X_0 = x, Y_0 = y.\n Define τ to be their first meet time: τ = min{n ≥ 0 : X_n = Y_n}.\n\n Show that ||P^n (x, ·) − P^n (y, ·)||_{TV} ≤ ℙ_{x,y} [τ > n].", "lean_statement": "theorem q321_chain_coupling {Ω' : Type*} [MeasurableSpace Ω'] (μ : Measure Ω') [IsProbabilityMeasure μ]\n {Ω : Type*} [Fintype Ω] [DecidableEq Ω] (P : Matrix Ω Ω ℝ) (hP : IsStochastic P)\n (x y : Ω) (Xc Yc : ℕ → Ω' → Ω) (τ : Ω' → ℕ) (n : ℕ)\n (hcoupX : HasMatrixMarginals μ Xc P x) (hcoupY : HasMatrixMarginals μ Yc P y)\n (hmeet : ∀ m ω, τ ω ≤ m → Xc m ω = Yc m ω) :\n tvDist (fun z => (P ^ n) x z) (fun z => (P ^ n) y z) ≤ tailProb μ τ n := by sorry", "true_to_source": "abstract", "proof": "private lemma isStochastic_pow_row_sum {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (n : ℕ) (x : Ω) :\n ∑ z, (P ^ n) x z = 1 := by\n induction n generalizing x with\n | zero => simp [pow_zero, Matrix.one_apply]\n | succ k ih =>\n rw [pow_succ]\n simp only [Matrix.mul_apply]\n rw [Finset.sum_comm]\n have hstep : ∀ w, ∑ z, (P ^ k) x w * P w z = (P ^ k) x w := by\n intro w\n rw [← Finset.mul_sum, hP.2 w, mul_one]\n simp_rw [hstep]\n exact ih x\n\nprivate lemma coupling_half {Ω' : Type*} [MeasurableSpace Ω'] (μ : Measure Ω')\n [IsProbabilityMeasure μ] {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (x y : Ω) (Xc Yc : ℕ → Ω' → Ω)\n (τ : Ω' → ℕ) (n : ℕ) (hcoupX : HasMatrixMarginals μ Xc P x)\n (hcoupY : HasMatrixMarginals μ Yc P y)\n (hmeet : ∀ m ω, τ ω ≤ m → Xc m ω = Yc m ω) (S : Finset Ω) :\n ∑ z ∈ S, ((P ^ n) x z - (P ^ n) y z) ≤ (μ {ω | n < τ ω}).toReal := by\n classical\n set T : Set Ω' := {ω | n < τ ω} with hT\n set U : Set Ω' := {ω | Xc n ω ∈ S} with hU\n set V : Set Ω' := {ω | Yc n ω ∈ S} with hV\n set W : Set Ω' := {ω | Xc n ω ∈ Sᶜ} with hW\n have hax : ∀ z, μ.real {ω | Xc n ω = z} = (P ^ n) x z := by\n intro z; rw [measureReal_def]; exact hcoupX n z\n have hby : ∀ z, μ.real {ω | Yc n ω = z} = (P ^ n) y z := by\n intro z; rw [measureReal_def]; exact hcoupY n z\n have hrow : ∑ z, (P ^ n) x z = 1 := isStochastic_pow_row_sum P hP n x\n have hUeq : U = ⋃ z ∈ S, {ω | Xc n ω = z} := by\n ext ω; simp only [hU, Set.mem_setOf_eq, Set.mem_iUnion, exists_prop]\n constructor\n · intro h; exact ⟨Xc n ω, h, rfl⟩\n · rintro ⟨z, hz, rfl⟩; exact hz\n have hWeq : W = ⋃ z ∈ Sᶜ, {ω | Xc n ω = z} := by\n ext ω; simp only [hW, Set.mem_setOf_eq, Set.mem_iUnion, exists_prop]\n constructor\n · intro h; exact ⟨Xc n ω, h, rfl⟩\n · rintro ⟨z, hz, rfl⟩; exact hz\n have hVeq : V = ⋃ z ∈ S, {ω | Yc n ω = z} := by\n ext ω; simp only [hV, Set.mem_setOf_eq, Set.mem_iUnion, exists_prop]\n constructor\n · intro h; exact ⟨Yc n ω, h, rfl⟩\n · rintro ⟨z, hz, rfl⟩; exact hz\n have hunivUW : (Set.univ : Set Ω') = U ∪ W := by\n ext ω\n simp only [hU, hW, Set.mem_univ, Set.mem_union, Set.mem_setOf_eq, true_iff]\n by_cases h : Xc n ω ∈ S\n · exact Or.inl h\n · exact Or.inr (Finset.mem_compl.mpr h)\n have huniv1 : μ.real (Set.univ : Set Ω') = 1 := by\n rw [measureReal_def, measure_univ]; simp\n have hUsub : μ.real U ≤ ∑ z ∈ S, (P ^ n) x z := by\n rw [hUeq]\n refine (measureReal_biUnion_finset_le S _).trans (le_of_eq ?_)\n exact Finset.sum_congr rfl (fun z _ => hax z)\n have hWsub : μ.real W ≤ ∑ z ∈ Sᶜ, (P ^ n) x z := by\n rw [hWeq]\n refine (measureReal_biUnion_finset_le Sᶜ _).trans (le_of_eq ?_)\n exact Finset.sum_congr rfl (fun z _ => hax z)\n have hUW : (1 : ℝ) ≤ μ.real U + μ.real W := by\n have h := measureReal_union_le (μ := μ) U W\n rw [← hunivUW, huniv1] at h\n exact h\n have hsplit_row : ∑ z ∈ S, (P ^ n) x z + ∑ z ∈ Sᶜ, (P ^ n) x z = 1 := by\n rw [Finset.sum_add_sum_compl]; exact hrow\n have hR1 : ∑ z ∈ S, (P ^ n) x z ≤ μ.real U := by linarith [hUW, hWsub, hsplit_row]\n have hR2 : μ.real V ≤ ∑ z ∈ S, (P ^ n) y z := by\n rw [hVeq]\n refine (measureReal_biUnion_finset_le S _).trans (le_of_eq ?_)\n exact Finset.sum_congr rfl (fun z _ => hby z)\n have hUTc : U ∩ Tᶜ = V ∩ Tᶜ := by\n ext ω\n simp only [hU, hV, hT, Set.mem_inter_iff, Set.mem_setOf_eq, Set.mem_compl_iff, not_lt]\n constructor\n · rintro ⟨hx, hτ⟩; exact ⟨by rw [← hmeet n ω hτ]; exact hx, hτ⟩\n · rintro ⟨hy, hτ⟩; exact ⟨by rw [hmeet n ω hτ]; exact hy, hτ⟩\n have hR3 : μ.real U - μ.real V ≤ μ.real T := by\n have hsplitU : (U ∩ T) ∪ (U ∩ Tᶜ) = U := by\n rw [← Set.inter_union_distrib_left, Set.union_compl_self, Set.inter_univ]\n have e1 : μ.real U ≤ μ.real (U ∩ T) + μ.real (U ∩ Tᶜ) := by\n calc μ.real U = μ.real ((U ∩ T) ∪ (U ∩ Tᶜ)) := by rw [hsplitU]\n _ ≤ μ.real (U ∩ T) + μ.real (U ∩ Tᶜ) := measureReal_union_le _ _\n have e2 : μ.real (U ∩ Tᶜ) = μ.real (V ∩ Tᶜ) := by rw [hUTc]\n have e3 : μ.real (V ∩ Tᶜ) ≤ μ.real V := measureReal_mono Set.inter_subset_left\n have e4 : μ.real (U ∩ T) ≤ μ.real T := measureReal_mono Set.inter_subset_right\n linarith [e1, e2, e3, e4]\n have hgoal : ∑ z ∈ S, ((P ^ n) x z - (P ^ n) y z) ≤ μ.real T := by\n rw [Finset.sum_sub_distrib]; linarith [hR1, hR2, hR3]\n rw [measureReal_def] at hgoal\n exact hgoal\n\ntheorem q321_chain_coupling {Ω' : Type*} [MeasurableSpace Ω'] (μ : Measure Ω') [IsProbabilityMeasure μ]\n {Ω : Type*} [Fintype Ω] [DecidableEq Ω] (P : Matrix Ω Ω ℝ) (hP : IsStochastic P)\n (x y : Ω) (Xc Yc : ℕ → Ω' → Ω) (τ : Ω' → ℕ) (n : ℕ)\n (hcoupX : HasMatrixMarginals μ Xc P x) (hcoupY : HasMatrixMarginals μ Yc P y)\n (hmeet : ∀ m ω, τ ω ≤ m → Xc m ω = Yc m ω) :\n tvDist (fun z => (P ^ n) x z) (fun z => (P ^ n) y z) ≤ tailProb μ τ n := by\n classical\n have key : ∑ z, |(P ^ n) x z - (P ^ n) y z| ≤ 2 * (μ {ω | n < τ ω}).toReal := by\n set S : Finset Ω := Finset.univ.filter (fun z => (P ^ n) y z ≤ (P ^ n) x z) with hS\n have hSeq : ∀ z ∈ S, |(P ^ n) x z - (P ^ n) y z| = (P ^ n) x z - (P ^ n) y z := by\n intro z hz\n rw [hS, Finset.mem_filter] at hz\n exact abs_of_nonneg (by linarith [hz.2])\n have hSceq : ∀ z ∈ Sᶜ, |(P ^ n) x z - (P ^ n) y z| = (P ^ n) y z - (P ^ n) x z := by\n intro z hz\n rw [Finset.mem_compl, hS, Finset.mem_filter] at hz\n simp only [Finset.mem_univ, true_and, not_le] at hz\n rw [abs_of_nonpos (by linarith)]; ring\n have h1 : ∑ z ∈ S, ((P ^ n) x z - (P ^ n) y z) ≤ (μ {ω | n < τ ω}).toReal :=\n coupling_half μ P hP x y Xc Yc τ n hcoupX hcoupY hmeet S\n have h2 : ∑ z ∈ Sᶜ, ((P ^ n) y z - (P ^ n) x z) ≤ (μ {ω | n < τ ω}).toReal :=\n coupling_half μ P hP y x Yc Xc τ n hcoupY hcoupX (fun m ω h => (hmeet m ω h).symm) Sᶜ\n calc ∑ z, |(P ^ n) x z - (P ^ n) y z|\n = ∑ z ∈ S, |(P ^ n) x z - (P ^ n) y z| + ∑ z ∈ Sᶜ, |(P ^ n) x z - (P ^ n) y z| :=\n (Finset.sum_add_sum_compl S _).symm\n _ = ∑ z ∈ S, ((P ^ n) x z - (P ^ n) y z) + ∑ z ∈ Sᶜ, ((P ^ n) y z - (P ^ n) x z) := by\n rw [Finset.sum_congr rfl hSeq, Finset.sum_congr rfl hSceq]\n _ ≤ (μ {ω | n < τ ω}).toReal + (μ {ω | n < τ ω}).toReal := add_le_add h1 h2\n _ = 2 * (μ {ω | n < τ ω}).toReal := by ring\n have e1 : tvDist (fun z => (P ^ n) x z) (fun z => (P ^ n) y z)\n = (1 / 2) * ∑ z, |(P ^ n) x z - (P ^ n) y z| := rfl\n have e2 : tailProb μ τ n = (μ {ω | n < τ ω}).toReal := rfl\n rw [e1, e2]\n linarith [key]", "time_seconds": 197, "topic": "Markov chains (finite & countable)" }, { "question_id": 323, "problem_name": "stochastic_processes_optional_stopping_bounded_process_super", "informal_statement": "Let (X_n) be a supermartingale that is uniformly bounded, i.e. |X_n| ≤ M almost surely for all n, and let τ be a stopping time.\n\nShow that E[X_τ] ≤ E[X_0].", "lean_statement": "theorem q323_optional_stopping_bounded_process_super (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ)\n (hX : Supermartingale X ℱ μ) (M : ℝ) (hbdd : ∀ n ω, |X n ω| ≤ M)\n (T : Ω → ℕ) (hT : IsStoppingTime ℱ (natStop T)) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ ≤ ∫ ω, X 0 ω ∂μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q323_optional_stopping_bounded_process_super (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ)\n (hX : Supermartingale X ℱ μ) (M : ℝ) (hbdd : ∀ n ω, |X n ω| ≤ M)\n (T : Ω → ℕ) (hT : IsStoppingTime ℱ (natStop T)) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ ≤ ∫ ω, X 0 ω ∂μ := by\n have hnYsub : Submartingale (stoppedProcess (-X) (natStop T)) ℱ μ :=\n hX.neg.stoppedProcess hT\n have hval : ∀ ω, stoppedValue X (natStop T) ω = X (T ω) ω := fun ω => rfl\n have hzero : ∀ ω, stoppedProcess X (natStop T) 0 ω = X 0 ω := by\n intro ω\n rw [MeasureTheory.stoppedProcess_eq_of_le\n (by simp only [natStop]; exact_mod_cast Nat.zero_le (T ω))]\n have heq0 : ∫ ω, stoppedProcess X (natStop T) 0 ω ∂μ = ∫ ω, X 0 ω ∂μ :=\n integral_congr_ae (ae_of_all μ hzero)\n have hle : ∀ n, ∫ ω, stoppedProcess X (natStop T) n ω ∂μ ≤ ∫ ω, X 0 ω ∂μ := by\n intro n\n have hraw := hnYsub.setIntegral_le (Nat.zero_le n) MeasurableSet.univ\n simp only [setIntegral_univ] at hraw\n have e0 : ∫ ω, stoppedProcess (-X) (natStop T) 0 ω ∂μ\n = -∫ ω, stoppedProcess X (natStop T) 0 ω ∂μ := by rw [← integral_neg]; rfl\n have en : ∫ ω, stoppedProcess (-X) (natStop T) n ω ∂μ\n = -∫ ω, stoppedProcess X (natStop T) n ω ∂μ := by rw [← integral_neg]; rfl\n rw [e0, en] at hraw\n linarith [heq0]\n have hmeas : ∀ n, AEStronglyMeasurable (fun ω => stoppedProcess X (natStop T) n ω) μ := by\n intro n\n have h : AEStronglyMeasurable (fun ω => -stoppedProcess X (natStop T) n ω) μ :=\n ((hnYsub.stronglyMeasurable n).mono (ℱ.le n)).aestronglyMeasurable\n exact (h.neg).congr (by filter_upwards with ω; simp)\n have hbound : ∀ n, ∀ᵐ ω ∂μ, ‖stoppedProcess X (natStop T) n ω‖ ≤ M :=\n fun n => ae_of_all μ fun ω => by rw [Real.norm_eq_abs]; exact hbdd _ ω\n have hlim : ∀ᵐ ω ∂μ, Tendsto (fun n => stoppedProcess X (natStop T) n ω) atTop\n (𝓝 (stoppedValue X (natStop T) ω)) :=\n ae_of_all μ fun ω => tendsto_atTop_of_eventually_const (i₀ := T ω) fun n hn => by\n have hle' : natStop T ω ≤ (↑n : WithTop ℕ) := by simp only [natStop]; exact_mod_cast hn\n rw [MeasureTheory.stoppedProcess_eq_of_ge (u := X) (i := n) hle', hval ω]; rfl\n have hdct := tendsto_integral_of_dominated_convergence (fun _ => M) hmeas\n (integrable_const M) hbound hlim\n exact le_of_tendsto' hdct hle", "time_seconds": 305, "topic": "Martingales & stopping" }, { "question_id": 325, "problem_name": "stochastic_processes_wald_martingale", "informal_statement": "Let X_1,X_2,… be i.i.d. with M(θ)=E[e^{θX}] < ∞ for some θ, and let S_n = Σ_{i=1}^n X_i.\n\nShow that Z_n = e^{θS_n} / M(θ)^n is a martingale.", "lean_statement": "theorem q325_wald_exponential_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (X : ℕ → Ω → ℝ) (θ M : ℝ) (hM : 0 < M)\n (hadap : Adapted ℱ (fun n ω => ∑ i ∈ Finset.range n, X i ω))\n (hint : ∀ n, Integrable (fun ω => Real.exp (θ * X n ω)) μ)\n (hSint : ∀ n, Integrable (fun ω => Real.exp (θ * (∑ i ∈ Finset.range n, X i ω))) μ)\n (hmgf : ∀ n, μ[fun ω => Real.exp (θ * X n ω) | ℱ n] =ᵐ[μ] fun _ => M) :\n Martingale (fun n ω => Real.exp (θ * (∑ i ∈ Finset.range n, X i ω)) / M ^ n) ℱ μ := by\n sorry", "true_to_source": "abstract", "proof": "theorem q325_wald_exponential_martingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (X : ℕ → Ω → ℝ) (θ M : ℝ) (hM : 0 < M)\n (hadap : Adapted ℱ (fun n ω => ∑ i ∈ Finset.range n, X i ω))\n (hint : ∀ n, Integrable (fun ω => Real.exp (θ * X n ω)) μ)\n (hSint : ∀ n, Integrable (fun ω => Real.exp (θ * (∑ i ∈ Finset.range n, X i ω))) μ)\n (hmgf : ∀ n, μ[fun ω => Real.exp (θ * X n ω) | ℱ n] =ᵐ[μ] fun _ => M) :\n Martingale (fun n ω => Real.exp (θ * (∑ i ∈ Finset.range n, X i ω)) / M ^ n) ℱ μ := by\n have hM0 : M ≠ 0 := ne_of_gt hM\n have hSM : ∀ n, StronglyMeasurable[ℱ n]\n (fun ω => Real.exp (θ * (∑ i ∈ Finset.range n, X i ω)) / M ^ n) := by\n intro n\n exact (Real.continuous_exp.comp_stronglyMeasurable\n ((hadap.stronglyAdapted n).const_mul θ)).div stronglyMeasurable_const\n refine martingale_nat (fun n => hSM n) (fun n => (hSint n).div_const (M ^ n)) ?_\n intro n\n have key : (fun ω => Real.exp (θ * (∑ i ∈ Finset.range (n+1), X i ω)) / M ^ (n+1))\n = (fun ω => Real.exp (θ * (∑ i ∈ Finset.range n, X i ω)) / M ^ (n+1))\n * (fun ω => Real.exp (θ * X n ω)) := by\n funext ω\n simp only [Pi.mul_apply, Finset.sum_range_succ, mul_add, Real.exp_add]\n ring\n have hfSM : StronglyMeasurable[ℱ n]\n (fun ω => Real.exp (θ * (∑ i ∈ Finset.range n, X i ω)) / M ^ (n+1)) :=\n (Real.continuous_exp.comp_stronglyMeasurable\n ((hadap.stronglyAdapted n).const_mul θ)).div stronglyMeasurable_const\n have hfg : Integrable ((fun ω => Real.exp (θ * (∑ i ∈ Finset.range n, X i ω)) / M ^ (n+1))\n * (fun ω => Real.exp (θ * X n ω))) μ := by\n rw [← key]; exact (hSint (n+1)).div_const (M ^ (n+1))\n have step : μ[(fun ω => Real.exp (θ * (∑ i ∈ Finset.range (n+1), X i ω)) / M ^ (n+1)) | ℱ n]\n =ᵐ[μ] fun ω => Real.exp (θ * (∑ i ∈ Finset.range n, X i ω)) / M ^ n := by\n rw [key]\n refine (condExp_mul_of_stronglyMeasurable_left hfSM hfg (hint n)).trans ?_\n refine (Filter.EventuallyEq.mul (Filter.EventuallyEq.refl _ _) (hmgf n)).trans ?_\n filter_upwards with ω\n simp only [Pi.mul_apply]\n rw [pow_succ]\n field_simp\n exact step.symm", "time_seconds": 375, "topic": "Martingales & stopping" }, { "question_id": 331, "problem_name": "stochastic_processes_delayed_slln_lemma", "informal_statement": "Let (X_i)_{i≥2} be i.i.d. with mean X̄ and E[|X|] < ∞, and let X_1 be a random variable independent of (X_i)_{i≥2} and finite with probability 1. Let S_n = Σ_{i=1}^n X_i.\n\nShow that lim_{n→∞} S_n/n = X̄ with probability 1.", "lean_statement": "theorem q331_delayed_slln (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ)\n (hindep : iIndepFun X μ)\n (hident : ∀ i, 1 ≤ i → IdentDistrib (X i) (X 1) μ μ)\n (hint : Integrable (X 1) μ)\n (xbar : ℝ) (hmean : ∫ ω, X 1 ω ∂μ = xbar)\n (S : ℕ → Ω → ℝ) (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, X i ω) :\n ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => S n ω / n) atTop (𝓝 xbar) := by\n sorry", "true_to_source": "literal", "proof": "theorem q331_delayed_slln (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ)\n (hindep : iIndepFun X μ)\n (hident : ∀ i, 1 ≤ i → IdentDistrib (X i) (X 1) μ μ)\n (hint : Integrable (X 1) μ)\n (xbar : ℝ) (hmean : ∫ ω, X 1 ω ∂μ = xbar)\n (S : ℕ → Ω → ℝ) (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, X i ω) :\n ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => S n ω / n) atTop (𝓝 xbar) := by\n set Y : ℕ → Ω → ℝ := fun i => X (i + 1) with hY\n have hintY : Integrable (Y 0) μ := hint\n have hidentY : ∀ i, IdentDistrib (Y i) (Y 0) μ μ := fun i => hident (i + 1) (by omega)\n have hlaw := strong_law_ae Y hintY (fun i j hij => hindep.indepFun (by omega)) hidentY\n have hμY : μ[Y 0] = xbar := hmean\n rw [hμY] at hlaw\n filter_upwards [hlaw] with ω hω\n have hTn : Tendsto (fun n : ℕ => (∑ i ∈ Finset.range n, Y i ω) / (n : ℝ)) atTop (𝓝 xbar) :=\n hω.congr (fun n => by rw [smul_eq_mul]; ring)\n -- ratio n/(n+1) → 1\n have hr0 : Tendsto (fun n : ℕ => 1 / ((n : ℝ) + 1)) atTop (𝓝 0) :=\n tendsto_one_div_add_atTop_nhds_zero_nat\n have hratio : Tendsto (fun n : ℕ => (n : ℝ) / ((n : ℝ) + 1)) atTop (𝓝 1) := by\n have h := (tendsto_const_nhds (x := (1 : ℝ))).sub hr0\n rw [sub_zero] at h\n refine h.congr' ?_\n filter_upwards [eventually_gt_atTop 0] with n hn\n have hne : (n : ℝ) + 1 ≠ 0 := by positivity\n field_simp\n ring\n -- (b) X 0 / (n+1) → 0\n have hb : Tendsto (fun n : ℕ => X 0 ω / ((n : ℝ) + 1)) atTop (𝓝 0) := by\n have h := hr0.const_mul (X 0 ω)\n rw [mul_zero] at h\n exact h.congr (fun n => by rw [mul_one_div])\n -- (a) (∑ Y)/(n+1) → xbar\n have ha : Tendsto (fun n : ℕ => (∑ i ∈ Finset.range n, Y i ω) / ((n : ℝ) + 1))\n atTop (𝓝 xbar) := by\n have hprod := hTn.mul hratio\n rw [mul_one] at hprod\n refine hprod.congr' ?_\n filter_upwards [eventually_gt_atTop 0] with n hn\n have hn0 : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (by omega)\n have hn1 : (n : ℝ) + 1 ≠ 0 := by positivity\n field_simp\n have hsum := ha.add hb\n rw [add_zero] at hsum\n refine (tendsto_add_atTop_iff_nat 1).mp ?_\n refine hsum.congr (fun n => ?_)\n show (∑ i ∈ Finset.range n, Y i ω) / ((n : ℝ) + 1) + X 0 ω / ((n : ℝ) + 1)\n = S (n + 1) ω / (((n + 1 : ℕ)) : ℝ)\n rw [hS (n + 1) ω, Finset.sum_range_succ']\n simp only [hY]\n push_cast\n ring", "time_seconds": 435, "topic": "Weak convergence & functional limits" }, { "question_id": 332, "problem_name": "stochastic_processes_gamblers_ruin_4", "informal_statement": "Consider a gambler betting on the outcome of a sequence of independent fair coin tosses.\n If head, he gains one dollar. If tail, he loses one dollar.\n If he reaches a fortune of N dollars, he stops. If his purse is ever empty, he stops.\n \n Show that the expected time for the gambler to stop is k(N-k).", "lean_statement": "theorem q332_gamblers_ruin_expected_time\n (N : ℕ) (hN : 0 < N) (e : ℕ → ℝ)\n (hb0 : e 0 = 0) (hbN : e N = 0)\n (hrec : ∀ k, 0 < k → k < N → e k = 1 + (e (k - 1) + e (k + 1)) / 2)\n (k : ℕ) (hk : k ≤ N) :\n e k = (k : ℝ) * ((N : ℝ) - (k : ℝ)) := by sorry", "true_to_source": "literal", "proof": "theorem q332_gamblers_ruin_expected_time\n (N : ℕ) (hN : 0 < N) (e : ℕ → ℝ)\n (hb0 : e 0 = 0) (hbN : e N = 0)\n (hrec : ∀ k, 0 < k → k < N → e k = 1 + (e (k - 1) + e (k + 1)) / 2)\n (k : ℕ) (hk : k ≤ N) :\n e k = (k : ℝ) * ((N : ℝ) - (k : ℝ)) := by\n -- General solution: e j = j*(e 1 + 1) - j^2.\n have key : ∀ j, j ≤ N → e j = (j : ℝ) * (e 1 + 1) - (j : ℝ) ^ 2 := by\n intro j\n induction j using Nat.strong_induction_on with\n | _ j ih =>\n intro hj\n match j with\n | 0 => simpa using hb0\n | 1 => push_cast; ring\n | (m + 2) =>\n have hk1 : 0 < m + 1 := Nat.succ_pos m\n have hk2 : m + 1 < N := by omega\n have hr := hrec (m + 1) hk1 hk2\n have e1 : e m = (m : ℝ) * (e 1 + 1) - (m : ℝ) ^ 2 := ih m (by omega) (by omega)\n have e2 : e (m + 1) = ((m : ℝ) + 1) * (e 1 + 1) - ((m : ℝ) + 1) ^ 2 := by\n have := ih (m + 1) (by omega) (by omega); push_cast at this; linarith\n have hstep : e (m + 1 + 1) = 2 * e (m + 1) - e m - 2 := by\n have hh : e (m + 1) = 1 + (e m + e (m + 1 + 1)) / 2 := by simpa using hr\n linarith\n rw [hstep, e1, e2]; push_cast; ring\n have hNpos : (N : ℝ) ≠ 0 := by exact_mod_cast hN.ne'\n have hslope : e 1 + 1 = (N : ℝ) := by\n have hkN := key N (le_refl N)\n rw [hbN] at hkN\n have hN2 : (N : ℝ) * (e 1 + 1) = (N : ℝ) ^ 2 := by linarith\n have := mul_left_cancel₀ hNpos (by rw [mul_comm]; rw [pow_two] at hN2; linarith :\n (N : ℝ) * (e 1 + 1) = (N : ℝ) * (N : ℝ))\n linarith [this]\n rw [key k hk, hslope]; ring", "time_seconds": 61, "topic": "Random walks & large deviations" }, { "question_id": 333, "problem_name": "stochastic_processes_two_state_stationary", "informal_statement": "For a two-state Markov chain with transition matrix [[1−a, a], [b, 1−b]] where 0 < a, b ≤ 1, the distribution π = (b/(a+b), a/(a+b)) is stationary.", "lean_statement": "theorem q333_two_state_stationary\n (a b : ℝ) (ha : 0 < a) (ha1 : a ≤ 1) (hb : 0 < b) (hb1 : b ≤ 1)\n (P : Matrix (Fin 2) (Fin 2) ℝ) (hP : P = !![1 - a, a; b, 1 - b])\n (π : Fin 2 → ℝ) (hπ : π = ![b / (a + b), a / (a + b)]) :\n IsStationary P π := by\n sorry", "true_to_source": "abstract", "proof": "theorem q333_two_state_stationary (a b : ℝ) (ha : 0 < a) (ha1 : a ≤ 1)\n (hb : 0 < b) (hb1 : b ≤ 1) (P : Matrix (Fin 2) (Fin 2) ℝ)\n (hP : P = !![1 - a, a; b, 1 - b]) (π : Fin 2 → ℝ)\n (hπ : π = ![b / (a + b), a / (a + b)]) : IsStationary P π := by\n have hab : (0 : ℝ) < a + b := by linarith\n subst hP hπ\n refine ⟨?_, ?_, ?_⟩\n · intro x\n fin_cases x <;> simp [Matrix.cons_val] <;> positivity\n · simp [Fin.sum_univ_two, Matrix.cons_val]\n field_simp\n ring\n · intro y\n fin_cases y <;>\n simp [Fin.sum_univ_two, Matrix.cons_val] <;>\n field_simp <;> ring", "time_seconds": 240, "topic": "Markov chains (finite & countable)" }, { "question_id": 335, "problem_name": "stochastic_processes_stationary_uniqueness_2", "informal_statement": "Suppose that P is irreducible.\n\n Show that π(x) = 1/(𝔼_x[τ_x^+]), ∀x ∈ Ω.", "lean_statement": "theorem q335_stationary_return_formula {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (hirr : IsIrreducible P)\n (g : Ω → Ω → ℝ) (hg : IsHittingSolution P g)\n (π : Ω → ℝ) (hπ : IsStationary P π) :\n ∀ x, π x = 1 / returnTime P g x := by sorry", "true_to_source": "abstract", "proof": "theorem q335_stationary_return_formula {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (hirr : IsIrreducible P)\n (g : Ω → Ω → ℝ) (hg : IsHittingSolution P g)\n (π : Ω → ℝ) (hπ : IsStationary P π) :\n ∀ x, π x = 1 / returnTime P g x := by\n obtain ⟨hπpos, hπsum, hπstat⟩ := hπ\n intro a\n -- Extended first-step relation: F z := 1 + ∑ w, P z w * g w a equals g z a off the\n -- diagonal and equals the return time at z = a.\n have hF : ∀ z, 1 + ∑ w, P z w * g w a\n = g z a + (if z = a then returnTime P g a else 0) := by\n intro z\n by_cases hz : z = a\n · rw [if_pos hz, hg z a, if_pos hz, zero_add, hz]; rfl\n · rw [if_neg hz, add_zero, hg z a, if_neg hz]\n -- Sum π z * F z, expanded via stationarity.\n have lhs_eq : (∑ z, π z * (1 + ∑ w, P z w * g w a)) = 1 + ∑ w, π w * g w a := by\n simp_rw [mul_add, mul_one, Finset.sum_add_distrib, hπsum]\n congr 1\n simp_rw [Finset.mul_sum]\n rw [Finset.sum_comm]\n apply Finset.sum_congr rfl\n intro w _\n simp_rw [← mul_assoc]\n rw [← Finset.sum_mul, hπstat w]\n -- Sum π z * F z, evaluated via the hitting identity.\n have rhs_eq : (∑ z, π z * (g z a + (if z = a then returnTime P g a else 0)))\n = (∑ w, π w * g w a) + π a * returnTime P g a := by\n simp_rw [mul_add, Finset.sum_add_distrib]\n congr 1\n simp_rw [mul_ite, mul_zero]\n rw [Finset.sum_ite_eq']\n simp\n have hsumF : (∑ z, π z * (1 + ∑ w, P z w * g w a))\n = (∑ z, π z * (g z a + (if z = a then returnTime P g a else 0))) := by\n apply Finset.sum_congr rfl\n intro z _\n rw [hF z]\n have key : π a * returnTime P g a = 1 := by\n have combined : 1 + ∑ w, π w * g w a\n = (∑ w, π w * g w a) + π a * returnTime P g a := by\n rw [← lhs_eq, ← rhs_eq]; exact hsumF\n linarith\n have hrt : returnTime P g a ≠ 0 := by\n intro h0; rw [h0, mul_zero] at key; exact zero_ne_one key\n rw [eq_div_iff hrt]\n exact key", "time_seconds": 122, "topic": "Markov chains (finite & countable)" }, { "question_id": 336, "problem_name": "stochastic_processes_ctmc_detailed_balance", "informal_statement": "Let (X(t)) be an irreducible continuous-time Markov chain with transition rates q_{ij}. Suppose there is a probability distribution (p_i) satisfying the detailed balance equations p_i q_{ij} = p_j q_{ji} for all i ≠ j.\n\nShow that (p_i) is the steady-state distribution, the process is reversible, and the embedded chain is positive-recurrent.", "lean_statement": "theorem q336_ctmc_detailed_balance {S : Type*} [Countable S]\n (ν : S → ℝ) (hν : ∀ i, 0 < ν i)\n (P : S → S → ℝ) (hP0 : ∀ i j, 0 ≤ P i j) (hPloop : ∀ i, P i i = 0)\n (hPstoch : ∀ i, ∑' j, P i j = 1)\n (q : S → S → ℝ) (hq : ∀ i j, q i j = ν i * P i j)\n (hνbal : ∀ i, ν i = ∑' k, q i k)\n (hirr : ∀ a b : S, ∃ (c : ℕ → S) (m : ℕ),\n c 0 = a ∧ c m = b ∧ ∀ t < m, 0 < P (c t) (c (t + 1)))\n (p : S → ℝ) (hp0 : ∀ i, 0 < p i) (hpsum : ∑' i, p i = 1)\n (hpν : Summable (fun j => p j * ν j))\n (hdb : ∀ i j, i ≠ j → p i * q i j = p j * q j i) :\n (∀ j, p j * ν j = ∑' i, p i * q i j) ∧\n (∀ i j, p i * q i j = p j * q j i) ∧\n (∃ π : S → ℝ, (∀ j, 0 ≤ π j) ∧ (∑' j, π j = 1) ∧\n (∀ j, ∑' i, π i * P i j = π j)) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q336_ctmc_detailed_balance {S : Type*} [Countable S]\n (ν : S → ℝ) (hν : ∀ i, 0 < ν i)\n (P : S → S → ℝ) (hP0 : ∀ i j, 0 ≤ P i j) (hPloop : ∀ i, P i i = 0)\n (hPstoch : ∀ i, ∑' j, P i j = 1)\n (q : S → S → ℝ) (hq : ∀ i j, q i j = ν i * P i j)\n (hνbal : ∀ i, ν i = ∑' k, q i k)\n (hirr : ∀ a b : S, ∃ (c : ℕ → S) (m : ℕ),\n c 0 = a ∧ c m = b ∧ ∀ t < m, 0 < P (c t) (c (t + 1)))\n (p : S → ℝ) (hp0 : ∀ i, 0 < p i) (hpsum : ∑' i, p i = 1)\n (hpν : Summable (fun j => p j * ν j))\n (hdb : ∀ i j, i ≠ j → p i * q i j = p j * q j i) :\n (∀ j, p j * ν j = ∑' i, p i * q i j) ∧\n (∀ i j, p i * q i j = p j * q j i) ∧\n (∃ π : S → ℝ, (∀ j, 0 ≤ π j) ∧ (∑' j, π j = 1) ∧\n (∀ j, ∑' i, π i * P i j = π j)) := by\n -- symmetric detailed balance for all pairs (including i = j)\n have hdball : ∀ i j, p i * q i j = p j * q j i := by\n intro i j\n rcases eq_or_ne i j with h | h\n · subst h; rfl\n · exact hdb i j h\n -- part 1\n have part1 : ∀ j, p j * ν j = ∑' i, p i * q i j := by\n intro j\n calc p j * ν j = p j * ∑' k, q j k := by rw [hνbal j]\n _ = ∑' k, p j * q j k := (tsum_mul_left).symm\n _ = ∑' k, p k * q k j := by\n exact tsum_congr (fun k => hdball j k)\n -- nonempty\n have hne : Nonempty S := by\n by_contra h\n rw [not_nonempty_iff] at h\n rw [tsum_empty] at hpsum\n exact one_ne_zero hpsum.symm\n obtain ⟨i0⟩ := hne\n set Z : ℝ := ∑' k, p k * ν k with hZdef\n have hZpos : 0 < Z :=\n Summable.tsum_pos hpν (fun k => le_of_lt (mul_pos (hp0 k) (hν k))) i0\n (mul_pos (hp0 i0) (hν i0))\n have hZne : Z ≠ 0 := ne_of_gt hZpos\n refine ⟨part1, hdball, ?_⟩\n refine ⟨fun j => p j * ν j / Z, ?_, ?_, ?_⟩\n · intro j\n exact div_nonneg (le_of_lt (mul_pos (hp0 j) (hν j))) (le_of_lt hZpos)\n · have : ∑' j, p j * ν j / Z = Z * Z⁻¹ := by\n simp only [div_eq_mul_inv]\n rw [tsum_mul_right]\n rw [this, mul_inv_cancel₀ hZne]\n · intro j\n have step : ∀ i, (p i * ν i / Z) * P i j = (p i * q i j) / Z := by\n intro i\n rw [hq i j]; ring\n rw [tsum_congr step]\n have : ∑' i, (p i * q i j) / Z = (∑' i, p i * q i j) / Z := by\n simp only [div_eq_mul_inv]\n rw [tsum_mul_right]\n rw [this, ← part1 j]", "time_seconds": 420, "topic": "Continuous-time Markov & queues" }, { "question_id": 338, "problem_name": "stochastic_processes_mmm_stationary", "informal_statement": "Consider an M/M/m queue with arrival rate λ, per-server service rate μ, and ρ = λ/(mμ) < 1.\n\nShow that the steady-state distribution satisfies p_i = p_0 (mρ)^i/i! for i ≤ m and p_i = p_0 ρ^i m^m/m! for i ≥ m.", "lean_statement": "theorem q338_mmm_stationary\n (m : ℕ) (hm : 0 < m)\n (lam mu : ℝ) (hlam : 0 < lam) (hmu : 0 < mu)\n (ρ : ℝ) (hρ : ρ = lam / ((m : ℝ) * mu)) (hρ1 : ρ < 1)\n (d : ℕ → ℝ) (hd : ∀ i, d i = ((min i m : ℕ) : ℝ) * mu)\n (p : ℕ → ℝ) (hp0 : ∀ i, 0 ≤ p i) (hpsum : ∑' i, p i = 1)\n (hbal0 : p 0 * lam = p 1 * d 1)\n (hbal : ∀ n, p (n + 1) * (lam + d (n + 1)) = p n * lam + p (n + 2) * d (n + 2)) :\n (∀ i, i ≤ m → p i = p 0 * ((m : ℝ) * ρ) ^ i / (Nat.factorial i : ℝ)) ∧\n (∀ i, m ≤ i → p i = p 0 * ρ ^ i * (m : ℝ) ^ m / (Nat.factorial m : ℝ)) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q338_mmm_stationary\n (m : ℕ) (hm : 0 < m)\n (lam mu : ℝ) (hlam : 0 < lam) (hmu : 0 < mu)\n (ρ : ℝ) (hρ : ρ = lam / ((m : ℝ) * mu)) (hρ1 : ρ < 1)\n (d : ℕ → ℝ) (hd : ∀ i, d i = ((min i m : ℕ) : ℝ) * mu)\n (p : ℕ → ℝ) (hp0 : ∀ i, 0 ≤ p i) (hpsum : ∑' i, p i = 1)\n (hbal0 : p 0 * lam = p 1 * d 1)\n (hbal : ∀ n, p (n + 1) * (lam + d (n + 1)) = p n * lam + p (n + 2) * d (n + 2)) :\n (∀ i, i ≤ m → p i = p 0 * ((m : ℝ) * ρ) ^ i / (Nat.factorial i : ℝ)) ∧\n (∀ i, m ≤ i → p i = p 0 * ρ ^ i * (m : ℝ) ^ m / (Nat.factorial m : ℝ)) := by\n have hm0 : (m : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hm.ne'\n have hmune : mu ≠ 0 := hmu.ne'\n -- detailed balance\n have hdb : ∀ n, p n * lam = p (n + 1) * d (n + 1) := by\n intro n\n induction n with\n | zero => exact hbal0\n | succ k ih =>\n have h := hbal k\n linear_combination h + ih\n have hlam_eq : lam = ρ * ((m : ℝ) * mu) := by\n have hmm : ((m : ℝ) * mu) ≠ 0 := mul_ne_zero hm0 hmune\n rw [hρ]; field_simp\n -- part 1\n have part1 : ∀ i, i ≤ m → p i = p 0 * ((m : ℝ) * ρ) ^ i / (Nat.factorial i : ℝ) := by\n intro i\n induction i with\n | zero => intro _; simp\n | succ k ih =>\n intro hk1\n have hk : k ≤ m := by omega\n have hpk := ih hk\n have hdk1 : d (k + 1) = ((k : ℝ) + 1) * mu := by\n rw [hd (k + 1), min_eq_left hk1]; push_cast; ring\n have hbalk := hdb k\n rw [hdk1] at hbalk\n have hk1ne : ((k : ℝ) + 1) ≠ 0 := by positivity\n have hkfac : (Nat.factorial k : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (Nat.factorial_pos k).ne'\n have hk1mu_ne : ((k : ℝ) + 1) * mu ≠ 0 := mul_ne_zero hk1ne hmune\n have hpk1 : p (k + 1) = p k * lam / (((k : ℝ) + 1) * mu) := by\n rw [eq_div_iff hk1mu_ne]; linarith [hbalk]\n have hfact : (Nat.factorial (k + 1) : ℝ) = ((k : ℝ) + 1) * (Nat.factorial k : ℝ) := by\n rw [Nat.factorial_succ]; push_cast; ring\n rw [hpk1, hpk, hfact, pow_succ, hlam_eq]\n field_simp\n refine ⟨part1, ?_⟩\n -- part 2\n intro i hi\n induction i, hi using Nat.le_induction with\n | base =>\n have h := part1 m le_rfl\n rw [h, mul_pow]; ring\n | succ k hk ih =>\n have hdk1 : d (k + 1) = (m : ℝ) * mu := by\n rw [hd (k + 1), min_eq_right (by omega : m ≤ k + 1)]\n have hbalk := hdb k\n rw [hdk1] at hbalk\n have hmmne : (m : ℝ) * mu ≠ 0 := mul_ne_zero hm0 hmune\n have hpk1 : p (k + 1) = p k * ρ := by\n have h2 : p (k + 1) * ((m : ℝ) * mu) = (p k * ρ) * ((m : ℝ) * mu) := by\n rw [mul_assoc, ← hlam_eq]; linarith [hbalk]\n exact mul_right_cancel₀ hmmne h2\n rw [hpk1, ih, pow_succ]; ring", "time_seconds": 338, "topic": "Continuous-time Markov & queues" }, { "question_id": 341, "problem_name": "stochastic_processes_branching_mean", "informal_statement": "Consider a branching process X_{n+1} = Σ_{k=1}^{X_n} Y_{k,n} where the offspring counts Y_{k,n} are i.i.d. with mean Ȳ.\n\nShow that E[X_n] = Ȳ^n E[X_0].", "lean_statement": "theorem q341_branching_mean (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℕ) (Y : ℕ → ℕ → Ω → ℕ) (ybar : ℝ)\n (hXint : ∀ n, Integrable (fun ω => (X n ω : ℝ)) μ)\n (hYint : ∀ k n, Integrable (fun ω => (Y k n ω : ℝ)) μ)\n (hYmean : ∀ k n, ∫ ω, (Y k n ω : ℝ) ∂μ = ybar)\n (hrec : ∀ n ω, X (n + 1) ω = ∑ k ∈ Finset.range (X n ω), Y k n ω)\n (hstop : ∀ n k : ℕ, IndepFun (fun ω => if k < X n ω then (1 : ℝ) else 0)\n (fun ω => (Y k n ω : ℝ)) μ) :\n ∀ n, ∫ ω, (X n ω : ℝ) ∂μ = ybar ^ n * ∫ ω, (X 0 ω : ℝ) ∂μ := by\n sorry", "true_to_source": "abstract", "proof": "theorem q341_branching_mean (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℕ) (Y : ℕ → ℕ → Ω → ℕ) (ybar : ℝ)\n (hXint : ∀ n, Integrable (fun ω => (X n ω : ℝ)) μ)\n (hYint : ∀ k n, Integrable (fun ω => (Y k n ω : ℝ)) μ)\n (hYmean : ∀ k n, ∫ ω, (Y k n ω : ℝ) ∂μ = ybar)\n (hrec : ∀ n ω, X (n + 1) ω = ∑ k ∈ Finset.range (X n ω), Y k n ω)\n (hstop : ∀ n k : ℕ, IndepFun (fun ω => if k < X n ω then (1 : ℝ) else 0)\n (fun ω => (Y k n ω : ℝ)) μ) :\n ∀ n, ∫ ω, (X n ω : ℝ) ∂μ = ybar ^ n * ∫ ω, (X 0 ω : ℝ) ∂μ := by\n have hstep : ∀ n, ∫ ω, (X (n+1) ω : ℝ) ∂μ = ybar * ∫ ω, (X n ω : ℝ) ∂μ := by\n intro n\n set ind : ℕ → Ω → ℝ := fun k ω => if k < X n ω then (1:ℝ) else 0 with hind\n set g : ℕ → Ω → ℝ := fun k ω => ind k ω * (Y k n ω : ℝ) with hg\n have hXaem : AEMeasurable (fun ω => (X n ω : ℝ)) μ := (hXint n).aemeasurable\n have hFk : ∀ k, Measurable (fun x : ℝ => if (k:ℝ) < x then (1:ℝ) else 0) := by\n intro k\n exact Measurable.ite (measurableSet_lt measurable_const measurable_id)\n measurable_const measurable_const\n have hind_asm : ∀ k, AEStronglyMeasurable (ind k) μ := by\n intro k\n have heq : ind k\n = (fun x : ℝ => if (k:ℝ) < x then (1:ℝ) else 0) ∘ (fun ω => (X n ω : ℝ)) := by\n funext ω\n simp only [hind, Function.comp_apply, Nat.cast_lt]\n rw [heq]\n exact ((hFk k).comp_aemeasurable hXaem).aestronglyMeasurable\n have hY_asm : ∀ k, AEStronglyMeasurable (fun ω => (Y k n ω : ℝ)) μ :=\n fun k => (hYint k n).aestronglyMeasurable\n have hg_asm : ∀ k, AEStronglyMeasurable (g k) μ := fun k => (hind_asm k).mul (hY_asm k)\n have hindsum : ∀ ω, ∑ k ∈ Finset.range (X n ω), ind k ω = (X n ω : ℝ) := by\n intro ω\n rw [Finset.sum_congr rfl\n (fun k hk => by simp only [hind]; rw [if_pos (Finset.mem_range.mp hk)])]\n simp\n have hgsum : ∀ ω, ∑ k ∈ Finset.range (X n ω), g k ω = (X (n+1) ω : ℝ) := by\n intro ω\n rw [hrec n ω]; push_cast\n apply Finset.sum_congr rfl\n intro k hk\n simp only [hg, hind, if_pos (Finset.mem_range.mp hk), one_mul]\n have P1 : ∀ ω, (X n ω : ℝ) = ∑' k, ind k ω := by\n intro ω\n rw [← hindsum ω, tsum_eq_sum (s := Finset.range (X n ω))\n (fun k hk => by simp only [hind]; rw [if_neg (by simpa using hk)])]\n have P2 : ∀ ω, (X (n+1) ω : ℝ) = ∑' k, g k ω := by\n intro ω\n rw [← hgsum ω, tsum_eq_sum (s := Finset.range (X n ω))\n (fun k hk => by simp only [hg, hind]; rw [if_neg (by simpa using hk)]; simp)]\n have hInt_g : ∀ k, ∫ ω, g k ω ∂μ = (∫ ω, ind k ω ∂μ) * ybar := by\n intro k\n have h := (hstop n k).integral_fun_mul_eq_mul_integral (hind_asm k) (hY_asm k)\n rw [hYmean k n] at h\n exact h\n have hsum_g : ∑' k, ∫⁻ ω, ‖g k ω‖ₑ ∂μ ≠ ∞ := by\n rw [← lintegral_tsum (fun k => (hg_asm k).enorm)]\n have hpt : (fun ω => ∑' k, ‖g k ω‖ₑ) =ᵐ[μ] (fun ω => ‖(X (n+1) ω : ℝ)‖ₑ) := by\n filter_upwards with ω\n have hnn : ∀ k, 0 ≤ g k ω := fun k => by\n simp only [hg, hind]; split <;> simp [Nat.cast_nonneg]\n calc ∑' k, ‖g k ω‖ₑ\n = ∑' k, ENNReal.ofReal (g k ω) := by\n apply tsum_congr; intro k; rw [Real.enorm_eq_ofReal (hnn k)]\n _ = ∑ k ∈ Finset.range (X n ω), ENNReal.ofReal (g k ω) := by\n apply tsum_eq_sum; intro k hk\n rw [show g k ω = 0 from by\n simp only [hg, hind]; rw [if_neg (by simpa using hk)]; simp]\n simp\n _ = ENNReal.ofReal (∑ k ∈ Finset.range (X n ω), g k ω) :=\n (ENNReal.ofReal_sum_of_nonneg (fun k _ => hnn k)).symm\n _ = ENNReal.ofReal (X (n+1) ω : ℝ) := by rw [hgsum ω]\n _ = ‖(X (n+1) ω : ℝ)‖ₑ := (Real.enorm_eq_ofReal (by positivity)).symm\n rw [lintegral_congr_ae hpt]\n exact (hXint (n+1)).2.ne\n have hsum_ind : ∑' k, ∫⁻ ω, ‖ind k ω‖ₑ ∂μ ≠ ∞ := by\n rw [← lintegral_tsum (fun k => (hind_asm k).enorm)]\n have hpt : (fun ω => ∑' k, ‖ind k ω‖ₑ) =ᵐ[μ] (fun ω => ‖(X n ω : ℝ)‖ₑ) := by\n filter_upwards with ω\n have hnn : ∀ k, 0 ≤ ind k ω := fun k => by simp only [hind]; split <;> simp\n calc ∑' k, ‖ind k ω‖ₑ\n = ∑' k, ENNReal.ofReal (ind k ω) := by\n apply tsum_congr; intro k; rw [Real.enorm_eq_ofReal (hnn k)]\n _ = ∑ k ∈ Finset.range (X n ω), ENNReal.ofReal (ind k ω) := by\n apply tsum_eq_sum; intro k hk\n rw [show ind k ω = 0 from by\n simp only [hind]; rw [if_neg (by simpa using hk)]]\n simp\n _ = ENNReal.ofReal (∑ k ∈ Finset.range (X n ω), ind k ω) :=\n (ENNReal.ofReal_sum_of_nonneg (fun k _ => hnn k)).symm\n _ = ENNReal.ofReal (X n ω : ℝ) := by rw [hindsum ω]\n _ = ‖(X n ω : ℝ)‖ₑ := (Real.enorm_eq_ofReal (by positivity)).symm\n rw [lintegral_congr_ae hpt]\n exact (hXint n).2.ne\n have hEXn1 : ∫ ω, (X (n+1) ω : ℝ) ∂μ = ∑' k, ∫ ω, g k ω ∂μ := by\n have e1 : ∫ ω, (X (n+1) ω : ℝ) ∂μ = ∫ ω, (∑' k, g k ω) ∂μ :=\n integral_congr_ae (ae_of_all _ P2)\n rw [e1, integral_tsum hg_asm hsum_g]\n have hEXn : ∫ ω, (X n ω : ℝ) ∂μ = ∑' k, ∫ ω, ind k ω ∂μ := by\n have e1 : ∫ ω, (X n ω : ℝ) ∂μ = ∫ ω, (∑' k, ind k ω) ∂μ :=\n integral_congr_ae (ae_of_all _ P1)\n rw [e1, integral_tsum hind_asm hsum_ind]\n rw [hEXn1]\n have hcongr : (∑' k, ∫ ω, g k ω ∂μ) = ∑' k, (∫ ω, ind k ω ∂μ) * ybar :=\n tsum_congr hInt_g\n rw [hcongr, tsum_mul_right, ← hEXn]\n ring\n intro n\n induction n with\n | zero => simp\n | succ n ih =>\n rw [hstep n, ih, pow_succ]\n ring", "time_seconds": 614, "topic": "Markov chains (finite & countable)" }, { "question_id": 344, "problem_name": "stochastic_processes_birth_death_stationary", "informal_statement": "Consider an irreducible birth-death Markov chain on {0,1,2,…} with up-probability p_i from state i, down-probability q_{i+1} from state i+1, and ρ_i = p_i/q_{i+1}.\n\nShow that a stationary distribution exists if and only if Σ_{i≥1} ∏_{j ∏ j ∈ Finset.range i, ρ j))\n ∧ (Summable (fun i => ∏ j ∈ Finset.range i, ρ j) →\n ∀ π : ℕ → ℝ, (∀ i, 0 ≤ π i) → (∑' i, π i = 1) →\n (∀ i, π i * p i = π (i + 1) * q (i + 1)) →\n ∀ i, π i = (∏ j ∈ Finset.range i, ρ j) / (∑' k, ∏ j ∈ Finset.range k, ρ j)) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q344_birth_death_stationary\n (p q : ℕ → ℝ) (hp : ∀ i, 0 < p i) (hq : ∀ i, 0 < q (i + 1))\n (ρ : ℕ → ℝ) (hρ : ∀ j, ρ j = p j / q (j + 1)) :\n ((∃ π : ℕ → ℝ, (∀ i, 0 ≤ π i) ∧ (∑' i, π i = 1) ∧\n (∀ i, π i * p i = π (i + 1) * q (i + 1)))\n ↔ Summable (fun i => ∏ j ∈ Finset.range i, ρ j))\n ∧ (Summable (fun i => ∏ j ∈ Finset.range i, ρ j) →\n ∀ π : ℕ → ℝ, (∀ i, 0 ≤ π i) → (∑' i, π i = 1) →\n (∀ i, π i * p i = π (i + 1) * q (i + 1)) →\n ∀ i, π i = (∏ j ∈ Finset.range i, ρ j) / (∑' k, ∏ j ∈ Finset.range k, ρ j)) := by\n have hρpos : ∀ j, 0 < ρ j := fun j => by rw [hρ j]; exact div_pos (hp j) (hq j)\n have hprodpos : ∀ i, 0 < ∏ j ∈ Finset.range i, ρ j :=\n fun i => Finset.prod_pos (fun j _ => hρpos j)\n have hrec : ∀ (π : ℕ → ℝ), (∀ i, π i * p i = π (i + 1) * q (i + 1)) →\n ∀ i, π (i + 1) = π i * ρ i := by\n intro π hbal i\n have hq' : q (i + 1) ≠ 0 := (hq i).ne'\n rw [hρ i, ← mul_div_assoc, eq_div_iff hq']\n exact (hbal i).symm\n have hclosed : ∀ (π : ℕ → ℝ), (∀ i, π (i + 1) = π i * ρ i) →\n ∀ i, π i = π 0 * ∏ j ∈ Finset.range i, ρ j := by\n intro π hr i\n induction i with\n | zero => simp\n | succ n ih => rw [hr n, ih, Finset.prod_range_succ]; ring\n refine ⟨⟨fun hex => ?_, fun hsm => ?_⟩, ?_⟩\n · -- existence → summable\n obtain ⟨π, hπnn, hπsum, hbal⟩ := hex\n have hcf := hclosed π (hrec π hbal)\n have hsummπ : Summable π := by\n by_contra h\n rw [tsum_eq_zero_of_not_summable h] at hπsum\n exact one_ne_zero hπsum.symm\n have hπ0pos : 0 < π 0 := by\n rcases (hπnn 0).lt_or_eq with h | h\n · exact h\n · exfalso\n have hall : ∀ i, π i = 0 := by intro i; rw [hcf i, ← h]; ring\n have : ∑' i, π i = 0 := by rw [tsum_congr hall]; exact tsum_zero\n rw [this] at hπsum; exact one_ne_zero hπsum.symm\n refine (hsummπ.mul_left (π 0)⁻¹).congr (fun i => ?_)\n rw [hcf i, ← mul_assoc, inv_mul_cancel₀ hπ0pos.ne', one_mul]\n · -- summable → existence\n have hSpos : 0 < ∑' k, ∏ j ∈ Finset.range k, ρ j :=\n hsm.tsum_pos (fun k => (hprodpos k).le) 0 (hprodpos 0)\n refine ⟨fun i => (∏ j ∈ Finset.range i, ρ j) / (∑' k, ∏ j ∈ Finset.range k, ρ j),\n fun i => div_nonneg (hprodpos i).le hSpos.le, ?_, ?_⟩\n · simp_rw [div_eq_mul_inv]\n rw [tsum_mul_right]\n exact mul_inv_cancel₀ hSpos.ne'\n · intro i\n dsimp only\n rw [Finset.prod_range_succ, hρ i]\n field_simp [hSpos.ne', (hq i).ne']\n · -- formula\n intro hSummable π hπnn hπsum hbal i\n have hcf := hclosed π (hrec π hbal)\n have hSpos : 0 < ∑' k, ∏ j ∈ Finset.range k, ρ j :=\n hSummable.tsum_pos (fun k => (hprodpos k).le) 0 (hprodpos 0)\n have hps : π 0 * (∑' k, ∏ j ∈ Finset.range k, ρ j) = 1 := by\n have hcalc : ∑' i, π i = π 0 * (∑' k, ∏ j ∈ Finset.range k, ρ j) := by\n rw [tsum_congr hcf, tsum_mul_left]\n rw [hcalc] at hπsum; exact hπsum\n have hπ0 : π 0 = 1 / (∑' k, ∏ j ∈ Finset.range k, ρ j) :=\n (eq_div_iff hSpos.ne').mpr hps\n rw [hcf i, hπ0]\n ring", "time_seconds": 272, "topic": "Markov chains (finite & countable)" }, { "question_id": 345, "problem_name": "stochastic_processes_poisson_sum_independent_counts", "informal_statement": "Let (N(t)) be a Poisson process of rate λ and let 0 ≤ t_1 < t_2 < t_3.\n\nShow that N(t_2) − N(t_1) and N(t_3) − N(t_2) are independent Poisson random variables, and that their sum is Poisson with mean λ(t_3 − t_1).", "lean_statement": "theorem q345_poisson_increments_independent_sum (μ : Measure Ω) [IsProbabilityMeasure μ]\n (lam : ℝ≥0) (hlam : 0 < lam) (t1 t2 t3 : ℝ≥0)\n (h12 : t1 ≤ t2) (h23 : t2 ≤ t3)\n (N : ℝ≥0 → Ω → ℕ)\n (hincr : ∀ a b : ℝ≥0, a ≤ b →\n HasLaw (fun ω => N b ω - N a ω) (poissonMeasure (lam * (b - a))) μ)\n (hindep : IndepFun (fun ω => N t2 ω - N t1 ω) (fun ω => N t3 ω - N t2 ω) μ) :\n HasLaw (fun ω => (N t2 ω - N t1 ω) + (N t3 ω - N t2 ω))\n (poissonMeasure (lam * (t3 - t1))) μ := by\n sorry", "true_to_source": "abstract", "proof": "theorem q345_poisson_increments_independent_sum (μ : Measure Ω) [IsProbabilityMeasure μ]\n (lam : ℝ≥0) (hlam : 0 < lam) (t1 t2 t3 : ℝ≥0)\n (h12 : t1 ≤ t2) (h23 : t2 ≤ t3)\n (N : ℝ≥0 → Ω → ℕ)\n (hincr : ∀ a b : ℝ≥0, a ≤ b →\n HasLaw (fun ω => N b ω - N a ω) (poissonMeasure (lam * (b - a))) μ)\n (hindep : IndepFun (fun ω => N t2 ω - N t1 ω) (fun ω => N t3 ω - N t2 ω) μ) :\n HasLaw (fun ω => (N t2 ω - N t1 ω) + (N t3 ω - N t2 ω))\n (poissonMeasure (lam * (t3 - t1))) μ := by\n have hX := hincr t1 t2 h12\n have hY := hincr t2 t3 h23\n have key := hindep.hasLaw_add_poissonMeasure hX hY\n have hrate : lam * (t2 - t1) + lam * (t3 - t2) = lam * (t3 - t1) := by\n rw [← mul_add, add_comm (t2 - t1) (t3 - t2), tsub_add_tsub_cancel h23 h12]\n rw [hrate] at key\n exact key", "time_seconds": 265, "topic": "Poisson processes" }, { "question_id": 349, "problem_name": "stochastic_processes_kolmogorov_nonneg_martingale_inequality", "informal_statement": "Let (Z_n) be a nonnegative martingale.\n\nShow that for every a > 0, Pr[sup_{n ≥ 1} Z_n ≥ a] ≤ E[Z_1]/a.", "lean_statement": "theorem q349_kolmogorov_nonneg_martingale_inequality (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (Z : ℕ → Ω → ℝ) (hZ : Martingale Z ℱ μ)\n (hnn : ∀ n ω, 0 ≤ Z n ω) (a : ℝ) (ha : 0 < a) :\n (μ {ω | ∃ n, a ≤ Z n ω}).toReal ≤ (∫ ω, Z 0 ω ∂μ) / a := by\n sorry", "true_to_source": "literal", "proof": "theorem q349_kolmogorov_nonneg_martingale_inequality (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (Z : ℕ → Ω → ℝ) (hZ : Martingale Z ℱ μ)\n (hnn : ∀ n ω, 0 ≤ Z n ω) (a : ℝ) (ha : 0 < a) :\n (μ {ω | ∃ n, a ≤ Z n ω}).toReal ≤ (∫ ω, Z 0 ω ∂μ) / a := by\n have hrm : ∀ n ω, runningMax Z n ω\n = (Finset.range (n+1)).sup' (by simp) (fun k => Z k ω) := by\n intro n ω\n unfold runningMax\n rw [← Finset.sup'_univ_eq_ciSup]\n apply le_antisymm\n · apply Finset.sup'_le; intro k _\n apply Finset.le_sup' (fun k => Z k ω); simp [Finset.mem_range, k.isLt]\n · apply Finset.sup'_le; intro j hj\n rw [Finset.mem_range] at hj\n exact Finset.le_sup' (fun k : Fin (n+1) => Z (↑k) ω) (Finset.mem_univ ⟨j, hj⟩)\n have hcoe : ((a.toNNReal : ℝ≥0) : ℝ) = a := Real.coe_toNNReal a ha.le\n have hsub : Submartingale Z ℱ μ := hZ.submartingale\n have hnn' : (0 : ℕ → Ω → ℝ) ≤ Z := fun n ω => hnn n ω\n have hstep : ∀ n, (μ {ω | a ≤ runningMax Z n ω}).toReal ≤ (∫ ω, Z 0 ω ∂μ) / a := by\n intro n\n have key := MeasureTheory.maximal_ineq hsub hnn' (ε := a.toNNReal) n\n rw [hcoe] at key\n simp only [← hrm] at key\n have hI : 0 ≤ ∫ ω in {ω | a ≤ runningMax Z n ω}, Z n ω ∂μ :=\n integral_nonneg (fun ω => hnn n ω)\n have key2 := ENNReal.toReal_mono ENNReal.ofReal_ne_top key\n rw [ENNReal.toReal_ofReal hI, ENNReal.toReal_mul, ENNReal.coe_toReal, hcoe] at key2\n have hsubint : ∫ ω in {ω | a ≤ runningMax Z n ω}, Z n ω ∂μ ≤ ∫ ω, Z n ω ∂μ :=\n setIntegral_le_integral (hsub.integrable n) (ae_of_all _ (fun ω => hnn n ω))\n have hEn : ∫ ω, Z n ω ∂μ = ∫ ω, Z 0 ω ∂μ := by\n have := hZ.setIntegral_eq (i := 0) (j := n) (Nat.zero_le n) (s := Set.univ)\n MeasurableSet.univ\n simpa using this.symm\n rw [hEn] at hsubint\n have hfin : a * (μ {ω | a ≤ runningMax Z n ω}).toReal ≤ ∫ ω, Z 0 ω ∂μ := le_trans key2 hsubint\n rw [le_div_iff₀ ha, mul_comm]\n exact hfin\n have hUnion : {ω | ∃ n, a ≤ Z n ω} = ⋃ n, {ω | a ≤ runningMax Z n ω} := by\n ext ω\n simp only [Set.mem_setOf_eq, Set.mem_iUnion]\n constructor\n · rintro ⟨n, hn⟩\n refine ⟨n, ?_⟩\n rw [hrm n ω]\n exact le_trans hn\n (Finset.le_sup' (fun k => Z k ω) (Finset.mem_range.mpr (Nat.lt_succ_self n)))\n · rintro ⟨n, hn⟩\n rw [hrm n ω, Finset.le_sup'_iff] at hn\n obtain ⟨k, _, hk⟩ := hn\n exact ⟨k, hk⟩\n have hmono : Monotone (fun n => {ω | a ≤ runningMax Z n ω}) := by\n intro i j hij ω hω\n simp only [Set.mem_setOf_eq] at hω ⊢\n rw [hrm i ω, Finset.le_sup'_iff] at hω\n rw [hrm j ω, Finset.le_sup'_iff]\n obtain ⟨k, hk, hk2⟩ := hω\n exact ⟨k, Finset.mem_range.mpr\n (lt_of_lt_of_le (Finset.mem_range.mp hk) (Nat.succ_le_succ hij)), hk2⟩\n have htend := tendsto_measure_iUnion_atTop (μ := μ) hmono\n have hc0 : 0 ≤ (∫ ω, Z 0 ω ∂μ) / a :=\n div_nonneg (integral_nonneg (fun ω => hnn 0 ω)) ha.le\n have hbound : ∀ n, μ {ω | a ≤ runningMax Z n ω} ≤ ENNReal.ofReal ((∫ ω, Z 0 ω ∂μ)/a) := by\n intro n\n rw [← ENNReal.ofReal_toReal (measure_ne_top μ _)]\n exact ENNReal.ofReal_le_ofReal (hstep n)\n have hlim : μ (⋃ n, {ω | a ≤ runningMax Z n ω}) ≤ ENNReal.ofReal ((∫ ω, Z 0 ω ∂μ)/a) :=\n le_of_tendsto htend (Filter.Eventually.of_forall hbound)\n rw [hUnion]\n calc (μ (⋃ n, {ω | a ≤ runningMax Z n ω})).toReal\n ≤ (ENNReal.ofReal ((∫ ω, Z 0 ω ∂μ)/a)).toReal :=\n ENNReal.toReal_mono ENNReal.ofReal_ne_top hlim\n _ = (∫ ω, Z 0 ω ∂μ)/a := ENNReal.toReal_ofReal hc0", "time_seconds": 515, "topic": "Martingales & stopping" }, { "question_id": 350, "problem_name": "stochastic_processes_supermartingale_stopping_time_3", "informal_statement": "Let (X_n)_{n ≥ 0} be a supermartingale, and let S ≤ T be bounded stopping times.\n\n Show that 𝔼[X_T] ≤ 𝔼[X_S].", "lean_statement": "theorem q350_super_optional_eq (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Supermartingale X ℱ μ) (S T : Ω → ℕ)\n (hS : IsStoppingTime ℱ (natStop S))\n (hT : IsStoppingTime ℱ (natStop T))\n (hST : ∀ ω, S ω ≤ T ω) (N : ℕ) (hTbdd : ∀ ω, T ω ≤ N) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ\n ≤ ∫ ω, stoppedValue X (natStop S) ω ∂μ := by sorry", "true_to_source": "literal", "proof": "theorem q350_super_optional_eq (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Supermartingale X ℱ μ) (S T : Ω → ℕ)\n (hS : IsStoppingTime ℱ (natStop S))\n (hT : IsStoppingTime ℱ (natStop T))\n (hST : ∀ ω, S ω ≤ T ω) (N : ℕ) (hTbdd : ∀ ω, T ω ≤ N) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ\n ≤ ∫ ω, stoppedValue X (natStop S) ω ∂μ := by\n have hle : natStop S ≤ natStop T := fun ω => by\n unfold natStop; exact WithTop.coe_le_coe.mpr (hST ω)\n have hbd : ∀ ω, natStop T ω ≤ (N : WithTop ℕ) := fun ω => by\n unfold natStop; exact WithTop.coe_le_coe.mpr (hTbdd ω)\n have key := (hX.neg).expected_stoppedValue_mono hS hT hle hbd\n have e1 : μ[stoppedValue (-X) (natStop S)] = -μ[stoppedValue X (natStop S)] := by\n simp only [stoppedValue, Pi.neg_apply, integral_neg]\n have e2 : μ[stoppedValue (-X) (natStop T)] = -μ[stoppedValue X (natStop T)] := by\n simp only [stoppedValue, Pi.neg_apply, integral_neg]\n rw [e1, e2] at key\n linarith", "time_seconds": 986, "topic": "Martingales & stopping" }, { "question_id": 351, "problem_name": "stochastic_processes_time_reversal_1", "informal_statement": "Let (X_n) be an irreducible Markov chain with transition matrix P and stationary distribution π.\n Define P' to be P'(x, y) = π(y)P(y, x) / π(x).\n Let (X'_n) be a Markov chain with transition matrix P'.\n\n Show that π is also stationary for P'.", "lean_statement": "theorem q351_time_reversal_stationary {Ω : Type*} [Fintype Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (π : Ω → ℝ) (hπpos : ∀ x, 0 < π x)\n (hπ : IsStationary P π) (P' : Matrix Ω Ω ℝ) (hP' : IsReversal P π P') :\n IsStationary P' π := by sorry", "true_to_source": "literal", "proof": "theorem q351_time_reversal_stationary {Ω : Type*} [Fintype Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (π : Ω → ℝ) (hπpos : ∀ x, 0 < π x)\n (hπ : IsStationary P π) (P' : Matrix Ω Ω ℝ) (hP' : IsReversal P π P') :\n IsStationary P' π := by\n refine ⟨fun x => le_of_lt (hπpos x), hπ.2.1, ?_⟩\n intro y\n have hstep : ∀ x, π x * P' x y = π y * P y x := by\n intro x\n have hx : π x ≠ 0 := ne_of_gt (hπpos x)\n rw [hP' x y]\n field_simp\n rw [Finset.sum_congr rfl (fun x _ => hstep x), ← Finset.mul_sum, hP.2 y, mul_one]", "time_seconds": 100, "topic": "Markov chains (finite & countable)" }, { "question_id": 352, "problem_name": "stochastic_processes_convex_of_martingale_submartingale", "informal_statement": "Let (X_n) be a martingale and let φ be a convex function with E|φ(X_n)| < ∞ for all n.\n\nShow that (φ(X_n)) is a submartingale, and in particular that (|X_n|^p) is a submartingale for every p ≥ 1 for which it is integrable.", "lean_statement": "theorem q352_convex_of_martingale_submartingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (X : ℕ → Ω → ℝ) (hX : Martingale X ℱ μ) :\n (∀ φ : ℝ → ℝ, ConvexOn ℝ Set.univ φ → (∀ n, Integrable (fun ω => φ (X n ω)) μ) →\n Submartingale (fun n ω => φ (X n ω)) ℱ μ) ∧\n (∀ p : ℝ, 1 ≤ p → (∀ n, Integrable (fun ω => |X n ω| ^ p) μ) →\n Submartingale (fun n ω => |X n ω| ^ p) ℱ μ) := by\n sorry", "true_to_source": "literal", "proof": "theorem q352_convex_of_martingale_submartingale (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (X : ℕ → Ω → ℝ) (hX : Martingale X ℱ μ) :\n (∀ φ : ℝ → ℝ, ConvexOn ℝ Set.univ φ → (∀ n, Integrable (fun ω => φ (X n ω)) μ) →\n Submartingale (fun n ω => φ (X n ω)) ℱ μ) ∧\n (∀ p : ℝ, 1 ≤ p → (∀ n, Integrable (fun ω => |X n ω| ^ p) μ) →\n Submartingale (fun n ω => |X n ω| ^ p) ℱ μ) := by\n have main : ∀ φ : ℝ → ℝ, ConvexOn ℝ Set.univ φ → (∀ n, Integrable (fun ω => φ (X n ω)) μ) →\n Submartingale (fun n ω => φ (X n ω)) ℱ μ := by\n intro φ hφ hint\n have hcont : Continuous φ := hφ.locallyLipschitz.continuous\n refine submartingale_nat ?_ hint ?_\n · intro n\n exact hcont.comp_stronglyMeasurable (hX.stronglyAdapted n)\n · intro i\n have hjensen := hφ.map_condExp_le_of_finiteDimensional (ℱ.le i)\n (hX.integrable (i + 1)) (hint (i + 1))\n have hmg := hX.condExp_ae_eq (Nat.le_succ i)\n filter_upwards [hjensen, hmg] with ω h1 h2\n simp only [Function.comp_apply] at h1 h2 ⊢\n rw [h2] at h1\n exact h1\n refine ⟨main, ?_⟩\n intro p hp hint\n have hconv : ConvexOn ℝ Set.univ (fun x : ℝ => |x| ^ p) := by\n have hrange : (fun x : ℝ => |x|) '' Set.univ = Set.Ici 0 := by\n rw [Set.image_univ]\n ext y\n simp only [Set.mem_range, Set.mem_Ici]\n constructor\n · rintro ⟨x, rfl⟩; exact abs_nonneg x\n · intro hy; exact ⟨y, abs_of_nonneg hy⟩\n have hf : ConvexOn ℝ Set.univ (fun x : ℝ => |x|) := by\n have := convexOn_univ_norm (E := ℝ)\n simpa [Real.norm_eq_abs] using this\n have hg : ConvexOn ℝ ((fun x : ℝ => |x|) '' Set.univ) (fun t : ℝ => t ^ p) := by\n rw [hrange]; exact convexOn_rpow hp\n have hmono : MonotoneOn (fun t : ℝ => t ^ p) ((fun x : ℝ => |x|) '' Set.univ) := by\n rw [hrange]\n intro a ha b hb hab\n exact Real.rpow_le_rpow ha hab (by linarith)\n have := hg.comp hf hmono\n simpa [Function.comp] using this\n exact main _ hconv hint", "time_seconds": 351, "topic": "Martingales & stopping" }, { "question_id": 353, "problem_name": "stochastic_processes_poisson_process_superposition_2", "informal_statement": "Let (N^i)_{i ≥ 1} be a family of independent Poisson processes with respective\n positive intensities (λ_i)_{i ≥ 1}. \n\n Show that two distinct Poisson processes in this family have no points in common.", "lean_statement": "theorem q353_poisson_no_common (μ : Measure Ω) [IsProbabilityMeasure μ] (A B : ℕ → Ω → ℝ)\n (hAmeas : ∀ i, Measurable (A i)) (hBmeas : ∀ j, Measurable (B j))\n (hpair : ∀ i j, IndepFun (A i) (B j) μ) (hAcont : ∀ i x, μ {ω | A i ω = x} = 0)\n (hBcont : ∀ j x, μ {ω | B j ω = x} = 0) :\n μ {ω | ∃ i j, A i ω = B j ω} = 0 := by sorry", "true_to_source": "abstract", "proof": "theorem q353_poisson_no_common (μ : Measure Ω) [IsProbabilityMeasure μ] (A B : ℕ → Ω → ℝ)\n (hAmeas : ∀ i, Measurable (A i)) (hBmeas : ∀ j, Measurable (B j))\n (hpair : ∀ i j, IndepFun (A i) (B j) μ) (hAcont : ∀ i x, μ {ω | A i ω = x} = 0)\n (hBcont : ∀ j x, μ {ω | B j ω = x} = 0) :\n μ {ω | ∃ i j, A i ω = B j ω} = 0 := by\n have key : ∀ i j, μ {ω | A i ω = B j ω} = 0 := by\n intro i j\n haveI : IsProbabilityMeasure (Measure.map (A i) μ) :=\n Measure.isProbabilityMeasure_map (hAmeas i).aemeasurable\n haveI : IsProbabilityMeasure (Measure.map (B j) μ) :=\n Measure.isProbabilityMeasure_map (hBmeas j).aemeasurable\n have hf : Measurable (fun ω => (A i ω, B j ω)) := (hAmeas i).prodMk (hBmeas j)\n have hD : MeasurableSet {p : ℝ × ℝ | p.1 = p.2} :=\n measurableSet_eq_fun measurable_fst measurable_snd\n have hmap := (indepFun_iff_map_prod_eq_prod_map_map (hAmeas i).aemeasurable\n (hBmeas j).aemeasurable).mp (hpair i j)\n have hpre : {ω | A i ω = B j ω}\n = (fun ω => (A i ω, B j ω)) ⁻¹' {p : ℝ × ℝ | p.1 = p.2} := rfl\n rw [hpre, ← Measure.map_apply hf hD, hmap, Measure.prod_apply hD]\n have hinner : ∀ x : ℝ,\n (Measure.map (B j) μ) (Prod.mk x ⁻¹' {p : ℝ × ℝ | p.1 = p.2}) = 0 := by\n intro x\n have hset : (Prod.mk x ⁻¹' {p : ℝ × ℝ | p.1 = p.2}) = {x} := by\n ext y; simp [eq_comm]\n rw [hset, Measure.map_apply (hBmeas j) (measurableSet_singleton x)]\n exact hBcont j x\n simp_rw [hinner]\n simp\n have hset : {ω | ∃ i j, A i ω = B j ω} = ⋃ i, ⋃ j, {ω | A i ω = B j ω} := by\n ext ω; simp\n rw [hset]\n exact measure_iUnion_null (fun i => measure_iUnion_null (fun j => key i j))\n", "time_seconds": 214, "topic": "Poisson processes" }, { "question_id": 356, "problem_name": "stochastic_processes_ctmc_sampled_time_approximation", "informal_statement": "Let (X(t)) be a continuous-time Markov chain with bounded exit rates, and let δ satisfy δ ≤ 1/sup_i ν_i. Form the sampled-time chain with transition probabilities q_{ij} δ for i ≠ j and 1 − ν_i δ for i = j.\n\nShow that the sampled-time chain has the same steady-state distribution as the process.", "lean_statement": "theorem q356_ctmc_sampled_time_approximation {S : Type*} [Countable S]\n (ν : S → ℝ) (hν : ∀ i, 0 < ν i)\n (P : S → S → ℝ) (hP0 : ∀ i j, 0 ≤ P i j) (hPstoch : ∀ i, ∑' j, P i j = 1)\n (hPloop : ∀ i, P i i = 0)\n (q : S → S → ℝ) (hq : ∀ i j, q i j = ν i * P i j)\n (νstar : ℝ) (hνstar0 : 0 < νstar) (hνstar : ∀ i, ν i ≤ νstar)\n (δ : ℝ) (hδ0 : 0 < δ) (hδ : δ ≤ 1 / νstar)\n (Pstar : S → S → ℝ)\n (hPstarDiag : ∀ i, Pstar i i = 1 - ν i * δ)\n (hPstarOff : ∀ i j, i ≠ j → Pstar i j = q i j * δ)\n (p : S → ℝ) (hpnn : ∀ j, 0 ≤ p j) (hpsum : ∑' j, p j = 1)\n (hbalance : ∀ j, p j * ν j = ∑' i, p i * q i j) :\n ∀ j, ∑' i, p i * Pstar i j = p j := by\n sorry", "true_to_source": "abstract", "proof": "theorem q356_ctmc_sampled_time_approximation {S : Type*} [Countable S]\n (ν : S → ℝ) (hν : ∀ i, 0 < ν i)\n (P : S → S → ℝ) (hP0 : ∀ i j, 0 ≤ P i j) (hPstoch : ∀ i, ∑' j, P i j = 1)\n (hPloop : ∀ i, P i i = 0)\n (q : S → S → ℝ) (hq : ∀ i j, q i j = ν i * P i j)\n (νstar : ℝ) (hνstar0 : 0 < νstar) (hνstar : ∀ i, ν i ≤ νstar)\n (δ : ℝ) (hδ0 : 0 < δ) (hδ : δ ≤ 1 / νstar)\n (Pstar : S → S → ℝ)\n (hPstarDiag : ∀ i, Pstar i i = 1 - ν i * δ)\n (hPstarOff : ∀ i j, i ≠ j → Pstar i j = q i j * δ)\n (p : S → ℝ) (hpnn : ∀ j, 0 ≤ p j) (hpsum : ∑' j, p j = 1)\n (hbalance : ∀ j, p j * ν j = ∑' i, p i * q i j) :\n ∀ j, ∑' i, p i * Pstar i j = p j := by\n classical\n have hsummp : Summable p := by\n by_contra h\n have h0 := tsum_eq_zero_of_not_summable h\n rw [h0] at hpsum\n exact one_ne_zero hpsum.symm\n have hsummP : ∀ i, Summable (fun k => P i k) := fun i => by\n by_contra h\n have h0 : ∑' k, P i k = 0 := tsum_eq_zero_of_not_summable h\n rw [hPstoch i] at h0\n exact one_ne_zero h0\n have hPle : ∀ i k, P i k ≤ 1 := fun i k => by\n calc P i k ≤ ∑' k', P i k' := (hsummP i).le_tsum k (fun k' _ => hP0 i k')\n _ = 1 := hPstoch i\n intro j\n have hg_nonneg : ∀ i, 0 ≤ p i * q i j := by\n intro i\n rw [hq]\n exact mul_nonneg (hpnn i) (mul_nonneg (hν i).le (hP0 i j))\n have hg_le : ∀ i, p i * q i j ≤ p i * νstar := by\n intro i\n rw [hq]\n apply mul_le_mul_of_nonneg_left _ (hpnn i)\n calc ν i * P i j ≤ νstar * 1 :=\n mul_le_mul (hνstar i) (hPle i j) (hP0 i j) hνstar0.le\n _ = νstar := mul_one νstar\n have hsummg : Summable (fun i => p i * q i j) :=\n Summable.of_nonneg_of_le hg_nonneg hg_le (hsummp.mul_right νstar)\n have hpoint : ∀ i, p i * Pstar i j\n = δ * (p i * q i j) + (if i = j then p j - δ * (p j * ν j) else 0) := by\n intro i\n by_cases hij : i = j\n · subst hij\n rw [hPstarDiag, if_pos rfl, hq, hPloop]\n ring\n · rw [hPstarOff i j hij, if_neg hij]\n ring\n calc ∑' i, p i * Pstar i j\n = ∑' i, (δ * (p i * q i j) + (if i = j then p j - δ * (p j * ν j) else 0)) :=\n tsum_congr hpoint\n _ = (∑' i, δ * (p i * q i j))\n + ∑' i, (if i = j then p j - δ * (p j * ν j) else 0) := by\n exact Summable.tsum_add (hsummg.mul_left δ) (hasSum_ite_eq j _).summable\n _ = δ * (∑' i, p i * q i j) + (p j - δ * (p j * ν j)) := by\n rw [tsum_mul_left, (hasSum_ite_eq j (p j - δ * (p j * ν j))).tsum_eq]\n _ = δ * (p j * ν j) + (p j - δ * (p j * ν j)) := by rw [← hbalance j]\n _ = p j := by ring", "time_seconds": 277, "topic": "Continuous-time Markov & queues" }, { "question_id": 359, "problem_name": "stochastic_processes_long_run", "informal_statement": "Let X_n represent the presence of Jellyfishes on the nth day in Tom's favorite Aussie beach,\n where we assume we have either jellyfishes present (1) or no jellyfishes found (2).\n \n The transition matrix is:\n 1 2\n 1 0.9 0.1\n 2 0.3 0.7\n \n Show that the long-run fraction of days where no jellyfishes were found is 0.25.", "lean_statement": "theorem q359_long_run\n (P : Matrix (Fin 2) (Fin 2) ℝ) (hP : P = !![0.9, 0.1; 0.3, 0.7])\n (π : Fin 2 → ℝ) (hπ : IsStationary P π) :\n π 1 = 0.25 := by sorry", "true_to_source": "literal", "proof": "theorem q359_long_run\n (P : Matrix (Fin 2) (Fin 2) ℝ) (hP : P = !![0.9, 0.1; 0.3, 0.7])\n (π : Fin 2 → ℝ) (hπ : IsStationary P π) :\n π 1 = 0.25 := by\n obtain ⟨hnn, hsum, hstat⟩ := hπ\n have h1 := hstat 1\n subst hP\n simp [Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons] at h1 hsum\n linarith", "time_seconds": 120, "topic": "Markov chains (finite & countable)" }, { "question_id": 361, "problem_name": "stochastic_processes_hitting_time_3", "informal_statement": "Suppose that the chain is irreducible with stationary measure π.\n Let τ_x be the hitting time: τ_x = min{n ≥ 0 : X_n = x}.\n Let t_⊙ = sum_x (𝔼_a [τ_x]π(x)).\n Let t_{hit} = max_{x,y} 𝔼_x [τ_y] ≥ t_⊙.\n\n Show that t_{hit} ≤ 2 max_w 𝔼_π [τ_w].", "lean_statement": "theorem q361_hitting_bound {Ω : Type*} [Fintype Ω] [Nonempty Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (hirr : IsIrreducible P)\n (g : Ω → Ω → ℝ) (hg : IsHittingSolution P g)\n (π : Ω → ℝ) (hπ : IsStationary P π) :\n (⨆ p : Ω × Ω, g p.1 p.2) ≤ 2 * ⨆ w, ∑ x, π x * g x w := by sorry", "true_to_source": "abstract", "proof": "namespace Q361Helpers\n\nopen Finset\n\n/-- Entries of powers of a nonnegative matrix are nonnegative. -/\ntheorem pow_nonneg_entry {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP1 : ∀ x y, 0 ≤ P x y) : ∀ n x y, 0 ≤ (P ^ n) x y := by\n intro n\n induction n with\n | zero => intro x y; simp [pow_zero, Matrix.one_apply]; split <;> norm_num\n | succ k ih =>\n intro x y\n rw [pow_succ, Matrix.mul_apply]\n apply Finset.sum_nonneg; intro z _\n exact mul_nonneg (ih x z) (hP1 z y)\n\n/-- Hitting times are nonnegative. -/\ntheorem g_nonneg {Ω : Type*} [Fintype Ω] [Nonempty Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP1 : ∀ x y, 0 ≤ P x y) (hP2 : ∀ x, ∑ y, P x y = 1)\n (g : Ω → Ω → ℝ) (hg : ∀ x y, g x y = if x = y then 0 else 1 + ∑ z, P x z * g z y)\n (y : Ω) : ∀ x, 0 ≤ g x y := by\n obtain ⟨x0, -, hx0⟩ := Finset.exists_min_image Finset.univ (fun x => g x y)\n ⟨Classical.arbitrary Ω, Finset.mem_univ _⟩\n have hmin : ∀ x, g x0 y ≤ g x y := fun x => hx0 x (Finset.mem_univ x)\n have hx0y : g x0 y = 0 := by\n by_cases h : x0 = y\n · rw [hg x0 y, if_pos h]\n · exfalso\n have hval := hg x0 y\n rw [if_neg h] at hval\n have hlb : (1 : ℝ) + ∑ z, P x0 z * g z y ≥ 1 + ∑ z, P x0 z * g x0 y := by\n have : (∑ z, P x0 z * g z y) ≥ ∑ z, P x0 z * g x0 y := by\n apply Finset.sum_le_sum; intro z _\n exact mul_le_mul_of_nonneg_left (hmin z) (hP1 x0 z)\n linarith\n have hsum : (∑ z, P x0 z * g x0 y) = g x0 y := by\n rw [← Finset.sum_mul, hP2 x0, one_mul]\n rw [hsum] at hlb\n linarith [hval, hlb]\n intro x\n calc 0 = g x0 y := hx0y.symm\n _ ≤ g x y := hmin x\n\n/-- Forward-closure reachability: a set closed under positive one-step transitions is\nclosed under positive `n`-step transitions. -/\ntheorem reach_closed {Ω : Type*} [Fintype Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP1 : ∀ x y, 0 ≤ P x y)\n (A : Ω → Prop) (hclosed : ∀ x, A x → ∀ z, 0 < P x z → A z)\n (x0 : Ω) (hx0 : A x0) : ∀ n z, 0 < (P ^ n) x0 z → A z := by\n have hpow := pow_nonneg_entry P hP1\n intro n\n induction n with\n | zero =>\n intro z hz\n rw [pow_zero, Matrix.one_apply] at hz\n by_cases h : x0 = z\n · rwa [← h]\n · simp [h] at hz\n | succ k ih =>\n intro z hz\n rw [pow_succ, Matrix.mul_apply] at hz\n have hex : ∃ w ∈ Finset.univ, (0:ℝ) < (P ^ k) x0 w * P w z := by\n apply Finset.exists_lt_of_sum_lt; simpa using hz\n obtain ⟨w, -, hw⟩ := hex\n have h1 : 0 < (P ^ k) x0 w :=\n lt_of_le_of_ne (hpow k x0 w) (fun h => by rw [← h, zero_mul] at hw; exact lt_irrefl _ hw)\n have h2 : 0 < P w z :=\n lt_of_le_of_ne (hP1 w z) (fun h => by rw [← h, mul_zero] at hw; exact lt_irrefl _ hw)\n exact hclosed w (ih w h1) z h2\n\n/-- Maximum-principle propagation step: at a maximizing point where `f` is harmonic, all\npositively-reachable neighbours also attain the maximum. -/\ntheorem prop_step {Ω : Type*} [Fintype Ω]\n (P : Matrix Ω Ω ℝ) (hP1 : ∀ x y, 0 ≤ P x y) (hP2 : ∀ x, ∑ y, P x y = 1)\n (f : Ω → ℝ) (m : ℝ) (hmax : ∀ z, f z ≤ m) (x : Ω) (hfx : f x = m)\n (hharm : f x = ∑ z, P x z * f z) (z : Ω) (hz : 0 < P x z) : f z = m := by\n have hsum0 : (∑ w, P x w * (m - f w)) = 0 := by\n have : (∑ w, P x w * (m - f w)) = (∑ w, P x w) * m - ∑ w, P x w * f w := by\n rw [Finset.sum_mul, ← Finset.sum_sub_distrib]\n apply Finset.sum_congr rfl; intro w _; ring\n rw [this, hP2 x, one_mul, ← hharm, hfx]; ring\n have hnonneg : ∀ w ∈ Finset.univ, 0 ≤ P x w * (m - f w) := by\n intro w _; exact mul_nonneg (hP1 x w) (by linarith [hmax w])\n have hzero := (Finset.sum_eq_zero_iff_of_nonneg hnonneg).mp hsum0 z (Finset.mem_univ z)\n have : m - f z = 0 := by\n rcases mul_eq_zero.mp hzero with h | h\n · linarith [hz]\n · exact h\n linarith\n\n/-- Return-time identity: `π w · (1 + ∑_z P(w,z) g(z,w)) = 1`. -/\ntheorem returnTime_identity {Ω : Type*} [Fintype Ω] [Nonempty Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (g : Ω → Ω → ℝ)\n (hg : ∀ x y, g x y = if x = y then 0 else 1 + ∑ z, P x z * g z y)\n (π : Ω → ℝ) (hpi2 : ∑ x, π x = 1)\n (hpi3 : ∀ y, ∑ x, π x * P x y = π y) (w : Ω) :\n π w * (1 + ∑ z, P w z * g z w) = 1 := by\n have hswap : (∑ x, π x * ∑ z, P x z * g z w) = ∑ x, π x * g x w := by\n have e1 : (∑ x, π x * ∑ z, P x z * g z w) = ∑ x, ∑ z, π x * (P x z * g z w) := by\n apply Finset.sum_congr rfl; intro x _; rw [Finset.mul_sum]\n rw [e1, Finset.sum_comm]\n apply Finset.sum_congr rfl; intro z _\n have : (∑ x, π x * (P x z * g z w)) = (∑ x, π x * P x z) * g z w := by\n rw [Finset.sum_mul]; apply Finset.sum_congr rfl; intro x _; ring\n rw [this, hpi3 z]\n have s1 : (∑ x, if x = w then 0 else π x) = 1 - π w := by\n have : (∑ x, if x = w then 0 else π x) = ∑ x, (π x - if x = w then π x else 0) := by\n apply Finset.sum_congr rfl; intro x _; by_cases hx : x = w <;> simp [hx]\n rw [this, Finset.sum_sub_distrib, hpi2, Finset.sum_ite_eq' Finset.univ w π]; simp\n have s2 : (∑ x, if x = w then 0 else π x * ∑ z, P x z * g z w)\n = (∑ x, π x * g x w) - π w * ∑ z, P w z * g z w := by\n have step1 : (∑ x, if x = w then 0 else π x * ∑ z, P x z * g z w)\n = (∑ x, π x * ∑ z, P x z * g z w) - π w * ∑ z, P w z * g z w := by\n have : (∑ x, if x = w then 0 else π x * ∑ z, P x z * g z w)\n = ∑ x, ((π x * ∑ z, P x z * g z w)\n - (if x = w then π w * ∑ z, P w z * g z w else 0)) := by\n apply Finset.sum_congr rfl; intro x _; by_cases hx : x = w <;> simp [hx]\n rw [this, Finset.sum_sub_distrib, Finset.sum_ite_eq' Finset.univ w]; simp\n rw [step1, hswap]\n have key : (∑ x, π x * g x w)\n = (1 - π w) + ((∑ x, π x * g x w) - π w * (∑ z, P w z * g z w)) := by\n have e1 : (∑ x, π x * g x w)\n = ∑ x, π x * (if x = w then 0 else 1 + ∑ z, P x z * g z w) := by\n apply Finset.sum_congr rfl; intro x _; rw [hg x w]\n have e2 : (∑ x, π x * (if x = w then 0 else 1 + ∑ z, P x z * g z w))\n = (∑ x, if x = w then 0 else π x)\n + (∑ x, if x = w then 0 else π x * ∑ z, P x z * g z w) := by\n rw [← Finset.sum_add_distrib]\n apply Finset.sum_congr rfl; intro x _\n by_cases hx : x = w <;> simp [hx] <;> ring\n conv_lhs => rw [e1, e2, s1, s2]\n nlinarith [key]\n\n/-- Kemeny function `K x = ∑_c π_c g(x,c)` is harmonic: `K x = ∑_z P(x,z) K(z)`. -/\ntheorem K_harmonic {Ω : Type*} [Fintype Ω] [Nonempty Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP1 : ∀ x y, 0 ≤ P x y) (hP2 : ∀ x, ∑ y, P x y = 1)\n (g : Ω → Ω → ℝ) (hg : ∀ x y, g x y = if x = y then 0 else 1 + ∑ z, P x z * g z y)\n (π : Ω → ℝ) (hpi1 : ∀ x, 0 ≤ π x) (hpi2 : ∑ x, π x = 1)\n (hRT : ∀ w, π w * (1 + ∑ z, P w z * g z w) = 1) (x : Ω) :\n (∑ c, π c * g x c) = ∑ z, P x z * (∑ c, π c * g z c) := by\n have hpx : π x * (1 + ∑ z, P x z * g z x) = 1 := hRT x\n have hpxpos : 0 < π x := by\n rcases (hpi1 x).lt_or_eq with h | h\n · exact h\n · exfalso; rw [← h, zero_mul] at hpx; norm_num at hpx\n have hret : π x * (∑ z, P x z * g z x) = 1 - π x := by nlinarith [hpx]\n have e1 : (∑ c, π c * g x c) = ∑ c, (if c = x then 0 else π c * (1 + ∑ z, P x z * g z c)) := by\n apply Finset.sum_congr rfl; intro c _\n rw [hg x c]; by_cases h : x = c\n · simp [h]\n · rw [if_neg h, if_neg (Ne.symm h)]\n have e2 : (∑ c, (if c = x then 0 else π c * (1 + ∑ z, P x z * g z c)))\n = (∑ c, if c = x then 0 else π c)\n + (∑ c, if c = x then 0 else π c * ∑ z, P x z * g z c) := by\n rw [← Finset.sum_add_distrib]; apply Finset.sum_congr rfl; intro c _\n by_cases h : c = x <;> simp [h] <;> ring\n have sa : (∑ c, if c = x then 0 else π c) = 1 - π x := by\n have : (∑ c, if c = x then 0 else π c) = ∑ c, (π c - if c = x then π c else 0) := by\n apply Finset.sum_congr rfl; intro c _; by_cases h : c = x <;> simp [h]\n rw [this, Finset.sum_sub_distrib, hpi2, Finset.sum_ite_eq' Finset.univ x π]; simp\n have sb : (∑ c, if c = x then 0 else π c * ∑ z, P x z * g z c)\n = (∑ z, P x z * (∑ c, π c * g z c)) - π x * (∑ z, P x z * g z x) := by\n have drop : (∑ c, if c = x then 0 else π c * ∑ z, P x z * g z c)\n = (∑ c, π c * ∑ z, P x z * g z c) - π x * ∑ z, P x z * g z x := by\n have : (∑ c, if c = x then 0 else π c * ∑ z, P x z * g z c)\n = ∑ c, ((π c * ∑ z, P x z * g z c)\n - (if c = x then π x * ∑ z, P x z * g z x else 0)) := by\n apply Finset.sum_congr rfl; intro c _; by_cases h : c = x <;> simp [h]\n rw [this, Finset.sum_sub_distrib, Finset.sum_ite_eq' Finset.univ x]; simp\n rw [drop]\n congr 1\n have l1 : (∑ c, π c * ∑ z, P x z * g z c) = ∑ c, ∑ z, π c * (P x z * g z c) := by\n apply Finset.sum_congr rfl; intro c _; rw [Finset.mul_sum]\n have l2 : (∑ z, P x z * (∑ c, π c * g z c)) = ∑ z, ∑ c, P x z * (π c * g z c) := by\n apply Finset.sum_congr rfl; intro z _; rw [Finset.mul_sum]\n rw [l1, l2, Finset.sum_comm]\n apply Finset.sum_congr rfl; intro z _; apply Finset.sum_congr rfl; intro c _; ring\n rw [e1, e2, sa, sb, hret]\n ring\n\n/-- The Kemeny function is constant on an irreducible chain. -/\ntheorem K_const {Ω : Type*} [Fintype Ω] [Nonempty Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP1 : ∀ x y, 0 ≤ P x y) (hP2 : ∀ x, ∑ y, P x y = 1)\n (hirr : ∀ x y, ∃ n : ℕ, 0 < (P ^ n) x y)\n (K : Ω → ℝ) (hharm : ∀ x, K x = ∑ z, P x z * K z) (a b : Ω) : K a = K b := by\n obtain ⟨x0, -, hx0⟩ := Finset.exists_max_image Finset.univ K\n ⟨Classical.arbitrary Ω, Finset.mem_univ _⟩\n set m := K x0 with hm\n have hmax : ∀ z, K z ≤ m := fun z => hx0 z (Finset.mem_univ z)\n have hAclosed : ∀ x, K x = m → ∀ z, 0 < P x z → K z = m := by\n intro x hx z hz\n exact prop_step P hP1 hP2 K m hmax x hx (hharm x) z hz\n have hall : ∀ y, K y = m := by\n intro y\n obtain ⟨n, hn⟩ := hirr x0 y\n exact reach_closed P hP1 (fun w => K w = m) hAclosed x0 rfl n y hn\n rw [hall a, hall b]\n\n/-- Hitting-time triangle inequality: `g a b ≤ g a c + g c b`. -/\ntheorem tri_ineq {Ω : Type*} [Fintype Ω] [Nonempty Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP1 : ∀ x y, 0 ≤ P x y) (hP2 : ∀ x, ∑ y, P x y = 1)\n (hirr : ∀ x y, ∃ n : ℕ, 0 < (P ^ n) x y)\n (g : Ω → Ω → ℝ) (hg : ∀ x y, g x y = if x = y then 0 else 1 + ∑ z, P x z * g z y)\n (gnn : ∀ x y, 0 ≤ g x y) (a b c : Ω) : g a b ≤ g a c + g c b := by\n by_cases hbc : b = c\n · subst hbc\n have h0 : g b b = 0 := by rw [hg b b, if_pos rfl]\n have := gnn a b; nlinarith [h0]\n set θ : Ω → ℝ := fun x => g x b - g x c with hθ\n have hharm : ∀ x, x ≠ b → x ≠ c → θ x = ∑ z, P x z * θ z := by\n intro x hxb hxc\n have gb : g x b = 1 + ∑ z, P x z * g z b := by rw [hg x b, if_neg hxb]\n have gc : g x c = 1 + ∑ z, P x z * g z c := by rw [hg x c, if_neg hxc]\n simp only [hθ]; rw [gb, gc]\n have : (∑ z, P x z * (g z b - g z c)) = (∑ z, P x z * g z b) - ∑ z, P x z * g z c := by\n rw [← Finset.sum_sub_distrib]; apply Finset.sum_congr rfl; intro z _; ring\n rw [this]; ring\n obtain ⟨x0, -, hx0⟩ := Finset.exists_max_image Finset.univ θ\n ⟨Classical.arbitrary Ω, Finset.mem_univ _⟩\n set m := θ x0 with hm\n have hmax : ∀ z, θ z ≤ m := fun z => hx0 z (Finset.mem_univ z)\n have hθc : θ c = g c b := by simp only [hθ]; rw [hg c c, if_pos rfl]; ring\n have hθb : θ b = - g b c := by simp only [hθ]; rw [hg b b, if_pos rfl]; ring\n have hmeq : m = θ c := by\n by_cases hcA : θ c = m\n · exact hcA.symm\n · by_cases hbA : θ b = m\n · exfalso\n have h1 : m ≤ 0 := by rw [← hbA, hθb]; linarith [gnn b c]\n have h2 : 0 ≤ θ c := by rw [hθc]; exact gnn c b\n have h3 : θ c ≤ m := hmax c\n apply hcA; linarith\n · exfalso\n have hAclosed : ∀ x, θ x = m → ∀ z, 0 < P x z → θ z = m := by\n intro x hx z hz\n have hxb : x ≠ b := fun h => hbA (h ▸ hx)\n have hxc : x ≠ c := fun h => hcA (h ▸ hx)\n exact prop_step P hP1 hP2 θ m hmax x hx (hharm x hxb hxc) z hz\n obtain ⟨n, hn⟩ := hirr x0 c\n exact hcA (reach_closed P hP1 (fun w => θ w = m) hAclosed x0 rfl n c hn)\n have hθa : θ a ≤ θ c := by rw [← hmeq]; exact hmax a\n rw [hθc] at hθa\n simp only [hθ] at hθa\n linarith\n\n/-- The core inequality: every hitting time is at most `2 * M`. -/\ntheorem core_bound {Ω : Type*} [Fintype Ω] [Nonempty Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP1 : ∀ x y, 0 ≤ P x y) (hP2 : ∀ x, ∑ y, P x y = 1)\n (hirr : ∀ x y, ∃ n : ℕ, 0 < (P ^ n) x y)\n (g : Ω → Ω → ℝ) (hg : ∀ x y, g x y = if x = y then 0 else 1 + ∑ z, P x z * g z y)\n (π : Ω → ℝ) (hpi1 : ∀ x, 0 ≤ π x) (hpi2 : ∑ x, π x = 1)\n (hpi3 : ∀ y, ∑ x, π x * P x y = π y)\n (M : ℝ) (hM : ∀ y : Ω, (∑ x, π x * g x y) ≤ M) (a b : Ω) :\n g a b ≤ 2 * M := by\n have gnn : ∀ x y, 0 ≤ g x y := fun x y => g_nonneg P hP1 hP2 g hg y x\n have hRT : ∀ w, π w * (1 + ∑ z, P w z * g z w) = 1 :=\n fun w => returnTime_identity P g hg π hpi2 hpi3 w\n have tri : ∀ x y z, g x y ≤ g x z + g z y :=\n fun x y z => tri_ineq P hP1 hP2 hirr g hg gnn x y z\n have Kconst : ∀ x y, (∑ c, π c * g x c) = (∑ c, π c * g y c) := by\n intro x y\n exact K_const P hP1 hP2 hirr (fun x => ∑ c, π c * g x c)\n (fun x => K_harmonic P hP1 hP2 g hg π hpi1 hpi2 hRT x) x y\n -- Step 1: g a b ≤ K a + Z b\n have hgab : g a b ≤ (∑ c, π c * g a c) + (∑ c, π c * g c b) := by\n have h1 : g a b = ∑ c, π c * g a b := by rw [← Finset.sum_mul, hpi2, one_mul]\n rw [h1, ← Finset.sum_add_distrib]\n apply Finset.sum_le_sum; intro c _\n have htri := tri a b c\n have : π c * g a b ≤ π c * (g a c + g c b) := mul_le_mul_of_nonneg_left htri (hpi1 c)\n nlinarith [this]\n -- Step 2: Z b ≤ M\n have hZb : (∑ c, π c * g c b) ≤ M := hM b\n -- Step 3: K a ≤ M\n have hKa : (∑ c, π c * g a c) ≤ M := by\n have hka_eq : (∑ c, π c * g a c) = ∑ x, π x * (∑ c, π c * g x c) := by\n have : (∑ x, π x * (∑ c, π c * g x c)) = ∑ x, π x * (∑ c, π c * g a c) := by\n apply Finset.sum_congr rfl; intro x _; rw [Kconst x a]\n rw [this, ← Finset.sum_mul, hpi2, one_mul]\n rw [hka_eq]\n have swap : (∑ x, π x * (∑ c, π c * g x c)) = ∑ c, π c * (∑ x, π x * g x c) := by\n have l1 : (∑ x, π x * (∑ c, π c * g x c)) = ∑ x, ∑ c, π x * (π c * g x c) := by\n apply Finset.sum_congr rfl; intro x _; rw [Finset.mul_sum]\n have l2 : (∑ c, π c * (∑ x, π x * g x c)) = ∑ c, ∑ x, π c * (π x * g x c) := by\n apply Finset.sum_congr rfl; intro c _; rw [Finset.mul_sum]\n rw [l1, l2, Finset.sum_comm]\n apply Finset.sum_congr rfl; intro c _; apply Finset.sum_congr rfl; intro x _; ring\n rw [swap]\n calc (∑ c, π c * (∑ x, π x * g x c)) ≤ ∑ c, π c * M := by\n apply Finset.sum_le_sum; intro c _\n exact mul_le_mul_of_nonneg_left (hM c) (hpi1 c)\n _ = M := by rw [← Finset.sum_mul, hpi2, one_mul]\n linarith\n\nend Q361Helpers\n\nopen Auto in\ntheorem q361_hitting_bound {Ω : Type*} [Fintype Ω] [Nonempty Ω] [DecidableEq Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (hirr : IsIrreducible P)\n (g : Ω → Ω → ℝ) (hg : IsHittingSolution P g)\n (π : Ω → ℝ) (hπ : IsStationary P π) :\n (⨆ p : Ω × Ω, g p.1 p.2) ≤ 2 * ⨆ w, ∑ x, π x * g x w := by\n obtain ⟨hP1, hP2⟩ := hP\n obtain ⟨hpi1, hpi2, hpi3⟩ := hπ\n have hreach : ∀ x y, ∃ n : ℕ, 0 < (P ^ n) x y := by\n intro x y\n obtain ⟨n, _, hn⟩ := hirr x y\n exact ⟨n, hn⟩\n set M := ⨆ w, ∑ x, π x * g x w with hMdef\n have hbddM : BddAbove (Set.range (fun w => ∑ x, π x * g x w)) := Finite.bddAbove_range _\n have hZle : ∀ y : Ω, (∑ x, π x * g x y) ≤ M := fun y => le_ciSup hbddM y\n apply ciSup_le\n intro p\n exact Q361Helpers.core_bound P hP1 hP2 hreach g hg π hpi1 hpi2 hpi3 M hZle p.1 p.2\n", "time_seconds": 1800, "time_seconds_is_lower_bound": true, "topic": "Markov chains (finite & countable)" }, { "question_id": 365, "problem_name": "stochastic_processes_gamblers_ruin_5", "informal_statement": "Let X = (X_n)_{n ≥ 0} be Gambler’s ruin with state space Ω = {0, 1, 2, ..., N}:\n X_0 = k, ℙ[X_{n + 1} = X_n + 1 | X_n] = 1/2, τ = min{n: X_n = 0 or N}.\n\n Show that that Y = (Yn := X_n^2 − n)_{n ≥ 0} is a martingale.", "lean_statement": "theorem q365_gambler_square_martingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hadap : Adapted ℱ X) (hint : ∀ n, Integrable (fun ω => (X n ω) ^ 2) μ)\n (hstep1 : ∀ n, ∀ᵐ ω ∂μ, |X (n + 1) ω - X n ω| = 1)\n (hmean : IsConstDrift μ ℱ X 0) :\n Martingale (fun n ω => (X n ω) ^ 2 - n) ℱ μ := by sorry", "true_to_source": "literal", "proof": "theorem q365_gambler_square_martingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hadap : Adapted ℱ X) (hint : ∀ n, Integrable (fun ω => (X n ω) ^ 2) μ)\n (hstep1 : ∀ n, ∀ᵐ ω ∂μ, |X (n + 1) ω - X n ω| = 1)\n (hmean : IsConstDrift μ ℱ X 0) :\n Martingale (fun n ω => (X n ω) ^ 2 - n) ℱ μ := by\n have hXsm : ∀ k, StronglyMeasurable[ℱ k] (X k) := fun k => (hadap k).stronglyMeasurable\n have hXmem : ∀ k, MemLp (X k) 2 μ := fun k =>\n (memLp_two_iff_integrable_sq ((hXsm k).mono (ℱ.le k)).aestronglyMeasurable).2 (hint k)\n refine martingale_nat ?_ ?_ ?_\n · -- StronglyAdapted\n intro n\n exact by\n simpa [pow_two] using ((hXsm n).mul (hXsm n)).sub (stronglyMeasurable_const)\n · -- Integrable (Y n)\n intro n\n exact (hint n).sub (integrable_const _)\n · -- one-step: Y n =ᵐ μ[Y (n+1) | ℱ n]\n intro n\n have hg_int : Integrable (fun ω => X (n+1) ω - X n ω) μ :=\n ((hXmem (n+1)).sub (hXmem n)).integrable (by norm_num)\n have hcg_int : Integrable\n ((fun ω => 2 * X n ω) * (fun ω => X (n+1) ω - X n ω)) μ :=\n ((hXmem n).const_mul 2).integrable_mul ((hXmem (n+1)).sub (hXmem n))\n have hYn_int : Integrable (fun ω => (X n ω) ^ 2 - (n:ℝ)) μ := (hint n).sub (integrable_const _)\n have hc_sm : StronglyMeasurable[ℱ n] (fun ω => 2 * X n ω) := (hXsm n).const_mul 2\n have hYn_sm : StronglyMeasurable[ℱ n] (fun ω => (X n ω) ^ 2 - (n:ℝ)) := by\n simpa [pow_two] using ((hXsm n).mul (hXsm n)).sub (stronglyMeasurable_const)\n -- a.e. algebraic identity\n have hidentity : (fun ω => (X (n+1) ω) ^ 2 - ((n:ℝ)+1)) =ᵐ[μ]\n (fun ω => 2 * X n ω) * (fun ω => X (n+1) ω - X n ω) + (fun ω => (X n ω) ^ 2 - (n:ℝ)) := by\n filter_upwards [hstep1 n] with ω hω\n have hsq : (X (n+1) ω - X n ω) ^ 2 = 1 := by\n have h := sq_abs (X (n+1) ω - X n ω)\n rw [hω] at h; simpa using h.symm\n simp only [Pi.add_apply, Pi.mul_apply]\n nlinarith [hsq]\n have e4 : μ[(fun ω => (X n ω) ^ 2 - (n:ℝ)) | ℱ n] =ᵐ[μ] (fun ω => (X n ω) ^ 2 - (n:ℝ)) := by\n rw [condExp_of_stronglyMeasurable (ℱ.le n) hYn_sm hYn_int]\n have step : μ[(fun n ω => (X n ω) ^ 2 - (n:ℝ)) (n+1) | ℱ n]\n =ᵐ[μ] (fun ω => (X n ω) ^ 2 - (n:ℝ)) := by\n calc μ[(fun n ω => (X n ω) ^ 2 - (n:ℝ)) (n+1) | ℱ n]\n =ᵐ[μ] μ[(fun ω => (X (n+1) ω) ^ 2 - ((n:ℝ)+1)) | ℱ n] := by\n apply condExp_congr_ae\n filter_upwards with ω\n push_cast; ring\n _ =ᵐ[μ] μ[(fun ω => 2 * X n ω) * (fun ω => X (n+1) ω - X n ω)\n + (fun ω => (X n ω) ^ 2 - (n:ℝ)) | ℱ n] := condExp_congr_ae hidentity\n _ =ᵐ[μ] μ[(fun ω => 2 * X n ω) * (fun ω => X (n+1) ω - X n ω) | ℱ n]\n + μ[(fun ω => (X n ω) ^ 2 - (n:ℝ)) | ℱ n] := condExp_add hcg_int hYn_int _\n _ =ᵐ[μ] (fun ω => 2 * X n ω) * μ[(fun ω => X (n+1) ω - X n ω) | ℱ n]\n + μ[(fun ω => (X n ω) ^ 2 - (n:ℝ)) | ℱ n] :=\n EventuallyEq.add (condExp_mul_of_stronglyMeasurable_left hc_sm hcg_int hg_int)\n EventuallyEq.rfl\n _ =ᵐ[μ] (fun ω => 2 * X n ω) * (fun _ => (0:ℝ))\n + (fun ω => (X n ω) ^ 2 - (n:ℝ)) :=\n EventuallyEq.add (EventuallyEq.mul EventuallyEq.rfl (hmean n)) e4\n _ =ᵐ[μ] (fun ω => (X n ω) ^ 2 - (n:ℝ)) := by\n filter_upwards with ω\n simp [Pi.add_apply, Pi.mul_apply]\n exact step.symm", "time_seconds": 156, "topic": "Martingales & stopping" }, { "question_id": 370, "problem_name": "stochastic_processes_reversed_chain_transition", "informal_statement": "Let (X_n) be an irreducible positive-recurrent Markov chain with stationary distribution π and transition matrix [P], and define P*_{ij} = π_j P_{ji}/π_i.\n\nShow that [P*] is a stochastic matrix and is the transition matrix of the time-reversed chain in steady state.", "lean_statement": "theorem q370_reversed_chain\n {S : Type*} [Countable S]\n (P : S → S → ℝ) (hP0 : ∀ i j, 0 ≤ P i j) (hPstoch : ∀ i, ∑' j, P i j = 1)\n (π : S → ℝ) (hπpos : ∀ i, 0 < π i) (hπsum : ∑' i, π i = 1)\n (hπstat : ∀ j, ∑' i, π i * P i j = π j)\n (Pstar : S → S → ℝ) (hPstar : ∀ i j, Pstar i j = π j * P j i / π i) :\n (∀ i j, 0 ≤ Pstar i j) ∧ (∀ i, ∑' j, Pstar i j = 1) ∧\n (∀ j, ∑' i, π i * Pstar i j = π j) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q370_reversed_chain\n {S : Type*} [Countable S]\n (P : S → S → ℝ) (hP0 : ∀ i j, 0 ≤ P i j) (hPstoch : ∀ i, ∑' j, P i j = 1)\n (π : S → ℝ) (hπpos : ∀ i, 0 < π i) (hπsum : ∑' i, π i = 1)\n (hπstat : ∀ j, ∑' i, π i * P i j = π j)\n (Pstar : S → S → ℝ) (hPstar : ∀ i j, Pstar i j = π j * P j i / π i) :\n (∀ i j, 0 ≤ Pstar i j) ∧ (∀ i, ∑' j, Pstar i j = 1) ∧\n (∀ j, ∑' i, π i * Pstar i j = π j) := by\n refine ⟨?_, ?_, ?_⟩\n · intro i j\n rw [hPstar]\n exact div_nonneg (mul_nonneg (hπpos j).le (hP0 j i)) (hπpos i).le\n · intro i\n have h1 : ∀ j, Pstar i j = (π j * P j i) * (π i)⁻¹ := by\n intro j; rw [hPstar]; ring\n rw [tsum_congr h1, tsum_mul_right, hπstat i]\n exact mul_inv_cancel₀ (hπpos i).ne'\n · intro j\n have h2 : ∀ i, π i * Pstar i j = π j * P j i := by\n intro i\n have hpi : π i ≠ 0 := (hπpos i).ne'\n rw [hPstar]; field_simp\n rw [tsum_congr h2, tsum_mul_left, hPstoch j, mul_one]", "time_seconds": 85, "topic": "Markov chains (finite & countable)" }, { "question_id": 372, "problem_name": "stochastic_processes_martingale_5", "informal_statement": "Let X = (X_n)_{n ≥ 0} be a martingale in L^2.\n\n Show that X is bounded in L^2 if and only if sum_{n ≥ 0} 𝔼[(X_{n + 1} − X_n)^2] < ∞.", "lean_statement": "theorem q372_l2_bounded (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hX : Martingale X ℱ μ) (hL2 : ∀ n, MemLp (X n) 2 μ) :\n (∃ C : ℝ, ∀ n, ∫ ω, (X n ω) ^ 2 ∂μ ≤ C) ↔\n Summable (fun n => ∫ ω, (X (n + 1) ω - X n ω) ^ 2 ∂μ) := by sorry", "true_to_source": "literal", "proof": "theorem q372_l2_bounded (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hX : Martingale X ℱ μ) (hL2 : ∀ n, MemLp (X n) 2 μ) :\n (∃ C : ℝ, ∀ n, ∫ ω, (X n ω) ^ 2 ∂μ ≤ C) ↔\n Summable (fun n => ∫ ω, (X (n + 1) ω - X n ω) ^ 2 ∂μ) := by\n haveI : SigmaFiniteFiltration μ ℱ := IsFiniteMeasure.sigmaFiniteFiltration μ ℱ\n have hInt : ∀ k, Integrable (X k) μ := fun k => (hL2 k).integrable (by norm_num)\n have hsq : ∀ k, Integrable (fun ω => (X k ω) ^ 2) μ := by\n intro k; simpa [pow_two] using (hL2 k).integrable_mul (hL2 k)\n -- Cross-term identity from the martingale property: ∫ X(n+1)·X n = ∫ X n².\n have cross : ∀ n, ∫ ω, X (n + 1) ω * X n ω ∂μ = ∫ ω, (X n ω) ^ 2 ∂μ := by\n intro n\n haveI : SigmaFinite (μ.trim (ℱ.le n)) := sigmaFinite_of_sigmaFiniteFiltration μ ℱ n\n have hXn_sm : StronglyMeasurable[ℱ n] (X n) := hX.stronglyMeasurable n\n have hprod_int : Integrable (fun ω => X (n + 1) ω * X n ω) μ :=\n (hL2 (n + 1)).integrable_mul (hL2 n)\n rw [show (∫ ω, X (n + 1) ω * X n ω ∂μ) = ∫ ω, ((X (n + 1)) * (X n)) ω ∂μ from rfl,\n ← integral_condExp (ℱ.le n) (f := (X (n + 1)) * (X n))]\n have hpull : μ[(X (n + 1)) * (X n) | ℱ n] =ᵐ[μ] μ[X (n + 1) | ℱ n] * (X n) :=\n condExp_mul_of_stronglyMeasurable_right hXn_sm hprod_int (hInt (n + 1))\n have hmart : μ[X (n + 1) | ℱ n] =ᵐ[μ] X n := hX.condExp_ae_eq (Nat.le_succ n)\n have hfin : μ[(X (n + 1)) * (X n) | ℱ n] =ᵐ[μ] fun ω => (X n ω) ^ 2 := by\n filter_upwards [hpull, hmart] with ω e1 e2\n simp only [Pi.mul_apply, e1, e2]; ring\n rw [integral_congr_ae hfin]\n -- Per-step Pythagoras: ∫ X(n+1)² = ∫ X n² + ∫ (X(n+1) - X n)².\n have step : ∀ n, ∫ ω, (X (n + 1) ω) ^ 2 ∂μ\n = ∫ ω, (X n ω) ^ 2 ∂μ + ∫ ω, (X (n + 1) ω - X n ω) ^ 2 ∂μ := by\n intro n\n have hcross_int : Integrable (fun ω => X (n + 1) ω * X n ω) μ :=\n (hL2 (n + 1)).integrable_mul (hL2 n)\n have e1 : ∫ ω, (X (n + 1) ω - X n ω) ^ 2 ∂μ\n = ∫ ω, ((X (n + 1) ω) ^ 2 - 2 * (X (n + 1) ω * X n ω) + (X n ω) ^ 2) ∂μ := by\n apply integral_congr_ae; filter_upwards with ω; ring\n rw [e1,\n integral_add (f := fun ω => (X (n + 1) ω) ^ 2 - 2 * (X (n + 1) ω * X n ω))\n (g := fun ω => (X n ω) ^ 2) ((hsq (n + 1)).sub (hcross_int.const_mul 2)) (hsq n),\n integral_sub (f := fun ω => (X (n + 1) ω) ^ 2) (g := fun ω => 2 * (X (n + 1) ω * X n ω))\n (hsq (n + 1)) (hcross_int.const_mul 2),\n integral_const_mul, cross]\n ring\n -- Telescoping: ∫ X n² = ∫ X 0² + ∑_{k ∫ ω, (X (k + 1) ω - X k ω) ^ 2 ∂μ with ha\n have sumeq : ∀ n, ∫ ω, (X n ω) ^ 2 ∂μ\n = (∫ ω, (X 0 ω) ^ 2 ∂μ) + ∑ k ∈ Finset.range n, a k := by\n intro n\n induction n with\n | zero => simp\n | succ m ih => rw [Finset.sum_range_succ, ← add_assoc, ← ih, step m]\n have aneg : ∀ k, 0 ≤ a k := fun k => integral_nonneg (fun ω => sq_nonneg _)\n have hS0 : 0 ≤ ∫ ω, (X 0 ω) ^ 2 ∂μ := integral_nonneg (fun ω => sq_nonneg _)\n constructor\n · -- Bounded second moments ⇒ bounded partial sums of nonneg terms ⇒ summable.\n rintro ⟨C, hC⟩\n apply summable_of_sum_range_le (c := C) aneg\n intro n\n linarith [hC n, sumeq n]\n · -- Summable ⇒ second moments bounded by ∫ X 0² + total sum.\n intro hsum\n refine ⟨(∫ ω, (X 0 ω) ^ 2 ∂μ) + ∑' k, a k, fun n => ?_⟩\n have hle : ∑ k ∈ Finset.range n, a k ≤ ∑' k, a k :=\n hsum.sum_le_tsum (Finset.range n) (fun i _ => aneg i)\n linarith [sumeq n]", "time_seconds": 157, "topic": "Martingales & stopping" }, { "question_id": 374, "problem_name": "stochastic_processes_supermartingale_stopping_time_4", "informal_statement": "Let (X_n)_{n ≥ 0} be a supermartingale, and let T be a stopping time which is finite a.s.\n Let there be an integrable random variable Y such that |Xn| ≤ Y for all n.\n\n Show that 𝔼[X_T] ≤ 𝔼[X_0].", "lean_statement": "theorem q374_super_dominated (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Supermartingale X ℱ μ) (T : Ω → ℕ)\n (hT : IsStoppingTime ℱ (natStop T)) (Y : Ω → ℝ) (hY : Integrable Y μ)\n (hdom : ∀ n, ∀ᵐ ω ∂μ, |X n ω| ≤ Y ω) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ ≤ ∫ ω, X 0 ω ∂μ := by sorry", "true_to_source": "literal", "proof": "theorem q374_super_dominated (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Supermartingale X ℱ μ) (T : Ω → ℕ)\n (hT : IsStoppingTime ℱ (natStop T)) (Y : Ω → ℝ) (hY : Integrable Y μ)\n (hdom : ∀ n, ∀ᵐ ω ∂μ, |X n ω| ≤ Y ω) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ ≤ ∫ ω, X 0 ω ∂μ := by\n set F : ℕ → Ω → ℝ := fun N ω => X (min (T ω) N) ω with hF\n have hπ : ∀ N, IsStoppingTime ℱ (natStop (fun ω => min (T ω) N)) := by\n intro N; have h := hT.min_const N; convert h using 2\n have hbdd : ∀ N, ∀ ω, natStop (fun ω => min (T ω) N) ω ≤ (N : ℕ∞) := by\n intro N ω; simp only [Auto.natStop]; exact Nat.cast_le.mpr (min_le_right _ _)\n have hsv : ∀ N, stoppedValue X (natStop (fun ω => min (T ω) N)) = F N := by\n intro N; ext ω; simp only [hF, stoppedValue, Auto.natStop]; rfl\n have hbound : ∀ N, ∫ ω, F N ω ∂μ ≤ ∫ ω, X 0 ω ∂μ := by\n intro N\n have hσ : IsStoppingTime ℱ (fun _ : Ω => ((0:ℕ) : ℕ∞)) := isStoppingTime_const ℱ 0\n have hle : (fun _ : Ω => ((0:ℕ) : ℕ∞)) ≤ natStop (fun ω => min (T ω) N) := by\n intro ω; simp [Auto.natStop]\n have h := hX.neg.expected_stoppedValue_mono hσ (hπ N) hle (hbdd N)\n have huntop : WithTop.untopA ((0:ℕ):ℕ∞) = 0 := rfl\n simp only [stoppedValue, Pi.neg_apply, integral_neg, huntop] at h\n have hrw : (fun a => X (natStop (fun ω => min (T ω) N) a).untopA a) = F N := by\n ext a; simp only [hF, Auto.natStop]; rfl\n rw [hrw] at h\n linarith [h]\n have hFint : ∀ N, Integrable (F N) μ := by\n intro N\n have := (hX.neg.integrable_stoppedValue (hπ N) (hbdd N)).neg\n rw [show (-stoppedValue (-X) (natStop (fun ω => min (T ω) N))) = F N by\n ext a; simp only [hF, stoppedValue, Pi.neg_apply, Auto.natStop, neg_neg]; rfl] at this\n exact this\n have h_bound : ∀ N, ∀ᵐ ω ∂μ, ‖F N ω‖ ≤ Y ω := by\n intro N\n have hall : ∀ᵐ ω ∂μ, ∀ k ∈ Finset.range (N+1), |X k ω| ≤ Y ω := by\n apply (ae_ball_iff (Finset.range (N+1)).countable_toSet).mpr\n intro k _; exact hdom k\n filter_upwards [hall] with ω hω\n have hmem : min (T ω) N ∈ Finset.range (N+1) := by\n simp only [Finset.mem_range]; omega\n simpa [hF, Real.norm_eq_abs] using hω _ hmem\n have h_lim : ∀ᵐ ω ∂μ, Tendsto (fun N => F N ω) atTop (nhds (stoppedValue X (natStop T) ω)) := by\n filter_upwards with ω\n have hv : stoppedValue X (natStop T) ω = X (T ω) ω := by\n simp only [stoppedValue, Auto.natStop]; rfl\n rw [hv]\n apply tendsto_atTop_of_eventually_const (i₀ := T ω)\n intro N hN; simp only [hF]; rw [min_eq_left hN]\n have htend := tendsto_integral_of_dominated_convergence Y\n (fun N => (hFint N).aestronglyMeasurable) hY h_bound h_lim\n exact le_of_tendsto' htend hbound", "time_seconds": 120, "topic": "Martingales & stopping" }, { "question_id": 379, "problem_name": "stochastic_processes_time_reversal_2", "informal_statement": "Let (X_n) be an irreducible Markov chain with transition matrix P and stationary distribution π.\n Define P' to be P'(x, y) = π(y)P(y, x) / π(x).\n Let (X'_n) be a Markov chain with transition matrix P'.\n\n Show that for any x_0, ..., x_n ∈ Ω, we have\n ℙ_π [X_0 = x_0, ..., X_n = x_n] = ℙ_π [X'_0 = x_n, ..., X'_n = x_0].", "lean_statement": "theorem q379_time_reversal_path {Ω : Type*} [Fintype Ω]\n (P : Matrix Ω Ω ℝ) (π : Ω → ℝ) (hπpos : ∀ x, 0 < π x) (hπ : IsStationary P π)\n (P' : Matrix Ω Ω ℝ) (hP' : IsReversal P π P') (n : ℕ) (x : ℕ → Ω) :\n π (x 0) * ∏ i ∈ Finset.range n, P (x i) (x (i + 1))\n = π (x n) * ∏ i ∈ Finset.range n, P' (x (i + 1)) (x i) := by sorry", "true_to_source": "literal", "proof": "theorem q379_time_reversal_path {Ω : Type*} [Fintype Ω]\n (P : Matrix Ω Ω ℝ) (π : Ω → ℝ) (hπpos : ∀ x, 0 < π x) (hπ : IsStationary P π)\n (P' : Matrix Ω Ω ℝ) (hP' : IsReversal P π P') (n : ℕ) (x : ℕ → Ω) :\n π (x 0) * ∏ i ∈ Finset.range n, P (x i) (x (i + 1))\n = π (x n) * ∏ i ∈ Finset.range n, P' (x (i + 1)) (x i) := by\n induction n with\n | zero => simp\n | succ k ih =>\n rw [Finset.prod_range_succ, Finset.prod_range_succ, hP' (x (k+1)) (x k)]\n have hk1 : π (x (k+1)) ≠ 0 := ne_of_gt (hπpos _)\n field_simp\n linear_combination P (x k) (x (k + 1)) * ih", "time_seconds": 187, "topic": "Markov chains (finite & countable)" }, { "question_id": 384, "problem_name": "stochastic_processes_stationary", "informal_statement": "Suppose a family holds a content for their three children.\n Let X_n represent the child that won the first place for the nth challenge,\n where the children are named Aaron, Bella, and Chad.\n After winning a challenge, the contestants are more confident in their abilities,\n and hence the probabilities they retain their first place on the following challenge are\n 0.7 for Aaron, 0.8 for Bella, and 0.9 for Chad. For those who did not win the prior challenge,\n the probability they win the next challenge is equal to the other contestants that did\n not win the prior challenge.\n \n Show that the winning percentages of each child stabilize.", "lean_statement": "theorem q384_stationary\n (P : Matrix (Fin 3) (Fin 3) ℝ)\n (hP : P = !![0.7, 0.15, 0.15; 0.1, 0.8, 0.1; 0.05, 0.05, 0.9]) :\n ∃ π : Fin 3 → ℝ, IsStationary P π := by sorry", "true_to_source": "abstract", "proof": "theorem q384_stationary\n (P : Matrix (Fin 3) (Fin 3) ℝ)\n (hP : P = !![0.7, 0.15, 0.15; 0.1, 0.8, 0.1; 0.05, 0.05, 0.9]) :\n ∃ π : Fin 3 → ℝ, IsStationary P π := by\n refine ⟨![2/11, 3/11, 6/11], ?_, ?_, ?_⟩\n · intro x; fin_cases x <;> norm_num\n · simp [Fin.sum_univ_three]; norm_num\n · intro y; subst hP\n fin_cases y <;>\n simp [Fin.sum_univ_three, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons] <;>\n norm_num", "time_seconds": 134, "topic": "Markov chains (finite & countable)" }, { "question_id": 385, "problem_name": "stochastic_processes_renewal_count_epoch_duality", "informal_statement": "Let (N(t)) be a renewal process with arrival epochs S_n.\n\nShow that the events satisfy {N(t) < n} = {S_n > t}, {N(t) ≥ n} = {S_n ≤ t}, and {N(t) > n} = {S_{n+1} ≤ t}.", "lean_statement": "theorem q385_renewal_count_epoch_duality (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hpos : ∀ i ω, 0 < X i ω)\n (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (S : ℕ → Ω → ℝ) (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, X i ω)\n (N : ℝ → Ω → ℕ)\n (hgreatest : ∀ (t : ℝ), 0 ≤ t → ∀ (ω : Ω), IsGreatest {n : ℕ | S n ω ≤ t} (N t ω)) :\n (∀ (n : ℕ) (t : ℝ), 0 ≤ t → {ω | N t ω < n} = {ω | t < S n ω}) ∧\n (∀ (n : ℕ) (t : ℝ), 0 ≤ t → {ω | n ≤ N t ω} = {ω | S n ω ≤ t}) ∧\n (∀ (n : ℕ) (t : ℝ), 0 ≤ t → {ω | n < N t ω} = {ω | S (n + 1) ω ≤ t}) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q385_renewal_count_epoch_duality (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hpos : ∀ i ω, 0 < X i ω)\n (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (S : ℕ → Ω → ℝ) (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, X i ω)\n (N : ℝ → Ω → ℕ)\n (hgreatest : ∀ (t : ℝ), 0 ≤ t → ∀ (ω : Ω), IsGreatest {n : ℕ | S n ω ≤ t} (N t ω)) :\n (∀ (n : ℕ) (t : ℝ), 0 ≤ t → {ω | N t ω < n} = {ω | t < S n ω}) ∧\n (∀ (n : ℕ) (t : ℝ), 0 ≤ t → {ω | n ≤ N t ω} = {ω | S n ω ≤ t}) ∧\n (∀ (n : ℕ) (t : ℝ), 0 ≤ t → {ω | n < N t ω} = {ω | S (n + 1) ω ≤ t}) := by\n have hmono : ∀ (ω : Ω) {a b : ℕ}, a ≤ b → S a ω ≤ S b ω := by\n intro ω a b hab\n rw [hS a ω, hS b ω]\n apply Finset.sum_le_sum_of_subset_of_nonneg\n · intro x hx\n simp only [Finset.mem_range] at *\n omega\n · intro i _ _\n exact (hpos i ω).le\n have eq2 : ∀ (n : ℕ) (t : ℝ), 0 ≤ t → {ω | n ≤ N t ω} = {ω | S n ω ≤ t} := by\n intro n t ht\n ext ω\n simp only [Set.mem_setOf_eq]\n constructor\n · intro h\n exact le_trans (hmono ω h) (hgreatest t ht ω).1\n · intro h\n exact (hgreatest t ht ω).2 h\n refine ⟨?_, eq2, ?_⟩\n · intro n t ht\n ext ω\n simp only [Set.mem_setOf_eq]\n constructor\n · intro h\n by_contra hc\n push_neg at hc\n have := (hgreatest t ht ω).2 hc\n omega\n · intro h\n by_contra hc\n push_neg at hc\n have : S n ω ≤ t := le_trans (hmono ω hc) (hgreatest t ht ω).1\n linarith\n · intro n t ht\n ext ω\n simp only [Set.mem_setOf_eq]\n constructor\n · intro h\n have hle : n + 1 ≤ N t ω := h\n exact le_trans (hmono ω hle) (hgreatest t ht ω).1\n · intro h\n have := (hgreatest t ht ω).2 h\n omega", "time_seconds": 116, "topic": "Renewal processes" }, { "question_id": 386, "problem_name": "stochastic_processes_absolute_value_submartingale", "informal_statement": "Let (Z_n) be a martingale.\n\nShow that (|Z_n|) is a submartingale.", "lean_statement": "theorem q386_abs_submartingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (Z : ℕ → Ω → ℝ) (hZ : Martingale Z ℱ μ) (hint : ∀ n, Integrable (fun ω => |Z n ω|) μ) :\n Submartingale (fun n ω => |Z n ω|) ℱ μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q386_abs_submartingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (Z : ℕ → Ω → ℝ) (hZ : Martingale Z ℱ μ) (hint : ∀ n, Integrable (fun ω => |Z n ω|) μ) :\n Submartingale (fun n ω => |Z n ω|) ℱ μ := by\n have hconv : ConvexOn ℝ Set.univ (fun x : ℝ => |x|) := by\n simpa [Real.norm_eq_abs] using (convexOn_univ_norm (E := ℝ))\n have hcont : Continuous (fun x : ℝ => |x|) := _root_.continuous_abs\n refine ⟨fun n => hcont.comp_stronglyMeasurable (hZ.1 n), fun i j hij => ?_, hint⟩\n have hjen := hconv.map_condExp_le_of_finiteDimensional (ℱ.le i) (hZ.integrable j) (hint j)\n have hmart := hZ.condExp_ae_eq hij\n filter_upwards [hjen, hmart] with ω hω hω2\n simp only [Function.comp_apply] at hω\n rw [hω2] at hω\n exact hω", "time_seconds": 73, "topic": "Martingales & stopping" }, { "question_id": 389, "problem_name": "stochastic_processes_square_submartingale", "informal_statement": "Let (Z_n) be a martingale with E[Z_n^2] < ∞ for all n.\n\nShow that (Z_n^2) is a submartingale.", "lean_statement": "theorem q389_sq_submartingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (Z : ℕ → Ω → ℝ) (hZ : Martingale Z ℱ μ)\n (hint : ∀ n, Integrable (fun ω => (Z n ω) ^ 2) μ) :\n Submartingale (fun n ω => (Z n ω) ^ 2) ℱ μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q389_sq_submartingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (Z : ℕ → Ω → ℝ) (hZ : Martingale Z ℱ μ)\n (hint : ∀ n, Integrable (fun ω => (Z n ω) ^ 2) μ) :\n Submartingale (fun n ω => (Z n ω) ^ 2) ℱ μ := by\n refine submartingale_nat (fun i => (hZ.stronglyAdapted i).pow 2) hint (fun i => ?_)\n have hmart : μ[Z (i + 1) | ℱ i] =ᵐ[μ] Z i := hZ.condExp_ae_eq (Nat.le_succ i)\n have hconv : ConvexOn ℝ Set.univ (fun x : ℝ => x ^ 2) := Even.convexOn_pow even_two\n have hcont : LowerSemicontinuous (fun x : ℝ => x ^ 2) := (continuous_pow 2).lowerSemicontinuous\n have hJensen := hconv.map_condExp_le_univ (ℱ.le i) hcont (hZ.integrable (i + 1)) (hint (i + 1))\n filter_upwards [hJensen, hmart] with ω h1 h2\n simpa [Function.comp_apply, h2] using h1", "time_seconds": 58, "topic": "Martingales & stopping" }, { "question_id": 391, "problem_name": "stochastic_processes_ctmc_chapman_kolmogorov", "informal_statement": "Let (X(t)) be a continuous-time Markov chain with transition functions P_{ij}(t).\n\nShow that for all s, t ≥ 0, P_{ij}(s+t) = Σ_k P_{ik}(s) P_{kj}(t).", "lean_statement": "theorem q391_ctmc_chapman_kolmogorov (μ : Measure Ω) [IsProbabilityMeasure μ]\n {S : Type*} [Countable S]\n (X : ℝ → Ω → S)\n (Pt : ℝ → S → S → ℝ)\n (hinit : ∀ i, 0 < (μ {ω | X 0 ω = i}).toReal)\n (hdef : ∀ (u : ℝ) (i j : S), 0 ≤ u →\n (μ {ω | X 0 ω = i ∧ X u ω = j}).toReal\n = Pt u i j * (μ {ω | X 0 ω = i}).toReal)\n (hmarkov : ∀ (s t : ℝ) (i k j : S), 0 ≤ s → 0 ≤ t →\n (μ {ω | X 0 ω = i ∧ X s ω = k ∧ X (s + t) ω = j}).toReal\n = Pt t k j * (μ {ω | X 0 ω = i ∧ X s ω = k}).toReal)\n (hpartition : ∀ (s t : ℝ) (i j : S), 0 ≤ s → 0 ≤ t →\n (μ {ω | X 0 ω = i ∧ X (s + t) ω = j}).toReal\n = ∑' k, (μ {ω | X 0 ω = i ∧ X s ω = k ∧ X (s + t) ω = j}).toReal) :\n ∀ (s t : ℝ) (i j : S), 0 ≤ s → 0 ≤ t →\n Pt (s + t) i j = ∑' k, Pt s i k * Pt t k j := by\n sorry", "true_to_source": "abstract", "proof": "theorem q391_ctmc_chapman_kolmogorov (μ : Measure Ω) [IsProbabilityMeasure μ]\n {S : Type*} [Countable S]\n (X : ℝ → Ω → S)\n (Pt : ℝ → S → S → ℝ)\n (hinit : ∀ i, 0 < (μ {ω | X 0 ω = i}).toReal)\n (hdef : ∀ (u : ℝ) (i j : S), 0 ≤ u →\n (μ {ω | X 0 ω = i ∧ X u ω = j}).toReal\n = Pt u i j * (μ {ω | X 0 ω = i}).toReal)\n (hmarkov : ∀ (s t : ℝ) (i k j : S), 0 ≤ s → 0 ≤ t →\n (μ {ω | X 0 ω = i ∧ X s ω = k ∧ X (s + t) ω = j}).toReal\n = Pt t k j * (μ {ω | X 0 ω = i ∧ X s ω = k}).toReal)\n (hpartition : ∀ (s t : ℝ) (i j : S), 0 ≤ s → 0 ≤ t →\n (μ {ω | X 0 ω = i ∧ X (s + t) ω = j}).toReal\n = ∑' k, (μ {ω | X 0 ω = i ∧ X s ω = k ∧ X (s + t) ω = j}).toReal) :\n ∀ (s t : ℝ) (i j : S), 0 ≤ s → 0 ≤ t →\n Pt (s + t) i j = ∑' k, Pt s i k * Pt t k j := by\n intro s t i j hs ht\n have ha : (μ {ω | X 0 ω = i}).toReal ≠ 0 := (hinit i).ne'\n have hst : (0:ℝ) ≤ s + t := by linarith\n have step : ∀ k, (μ {ω | X 0 ω = i ∧ X s ω = k ∧ X (s + t) ω = j}).toReal\n = (Pt s i k * Pt t k j) * (μ {ω | X 0 ω = i}).toReal := by\n intro k\n rw [hmarkov s t i k j hs ht, hdef s i k hs]; ring\n have key : Pt (s + t) i j * (μ {ω | X 0 ω = i}).toReal\n = (∑' k, Pt s i k * Pt t k j) * (μ {ω | X 0 ω = i}).toReal := by\n rw [← hdef (s + t) i j hst, hpartition s t i j hs ht, tsum_congr step, tsum_mul_right]\n exact mul_right_cancel₀ ha key", "time_seconds": 70, "topic": "Continuous-time Markov & queues" }, { "question_id": 402, "problem_name": "stochastic_processes_prokhorov_theorem", "informal_statement": "Let (P_n) be a tight sequence of probability measures on a Polish space.\n\nShow that (P_n) has a weakly convergent subsequence.", "lean_statement": "theorem q402_prokhorov_theorem\n {E : Type*} [MeasurableSpace E] [MetricSpace E] [PolishSpace E] [BorelSpace E]\n (Ps : ℕ → ProbabilityMeasure E)\n (htight : IsTightMeasureSet {μ : Measure E | ∃ n, μ = (Ps n : Measure E)}) :\n ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∃ P : ProbabilityMeasure E,\n Tendsto (fun n => Ps (φ n)) atTop (𝓝 P) := by\n sorry", "true_to_source": "literal", "proof": "theorem q402_prokhorov_theorem\n {E : Type*} [MeasurableSpace E] [MetricSpace E] [PolishSpace E] [BorelSpace E]\n (Ps : ℕ → ProbabilityMeasure E)\n (htight : IsTightMeasureSet {μ : Measure E | ∃ n, μ = (Ps n : Measure E)}) :\n ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∃ P : ProbabilityMeasure E,\n Tendsto (fun n => Ps (φ n)) atTop (𝓝 P) := by\n set S : Set (ProbabilityMeasure E) := Set.range Ps with hS\n have hset : {((μ : ProbabilityMeasure E) : Measure E) | μ ∈ S}\n = {μ : Measure E | ∃ n, μ = (Ps n : Measure E)} := by\n ext ν\n constructor\n · rintro ⟨μ, ⟨n, rfl⟩, rfl⟩; exact ⟨n, rfl⟩\n · rintro ⟨n, rfl⟩; exact ⟨Ps n, ⟨n, rfl⟩, rfl⟩\n have htight' : IsTightMeasureSet {((μ : ProbabilityMeasure E) : Measure E) | μ ∈ S} := by\n rw [hset]; exact htight\n have hcl : IsCompact (closure S) := isCompact_closure_of_isTightMeasureSet htight'\n have hmem : ∀ n, Ps n ∈ closure S := fun n => subset_closure ⟨n, rfl⟩\n obtain ⟨P, -, φ, hφ, hlim⟩ := hcl.tendsto_subseq hmem\n exact ⟨φ, hφ, P, hlim⟩", "time_seconds": 129, "topic": "Weak convergence & functional limits" }, { "question_id": 407, "problem_name": "stochastic_processes_martingale_convergence_theorem", "informal_statement": "Let (Z_n) be a martingale satisfying E[|Z_n|] ≤ M < ∞ for all n.\n\nShow that Z_n converges with probability 1 to a random variable Z with E[|Z|] < ∞.", "lean_statement": "theorem q407_martingale_convergence (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (Z : ℕ → Ω → ℝ) (hZ : Martingale Z ℱ μ)\n (M : ℝ) (hbdd : ∀ n, ∫ ω, |Z n ω| ∂μ ≤ M) :\n ∃ Zinf : Ω → ℝ, Integrable Zinf μ ∧\n ∀ᵐ ω ∂μ, Tendsto (fun n => Z n ω) atTop (𝓝 (Zinf ω)) := by\n sorry", "true_to_source": "literal", "proof": "theorem q407_martingale_convergence (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) (Z : ℕ → Ω → ℝ) (hZ : Martingale Z ℱ μ)\n (M : ℝ) (hbdd : ∀ n, ∫ ω, |Z n ω| ∂μ ≤ M) :\n ∃ Zinf : Ω → ℝ, Integrable Zinf μ ∧\n ∀ᵐ ω ∂μ, Tendsto (fun n => Z n ω) atTop (𝓝 (Zinf ω)) := by\n refine ⟨ℱ.limitProcess Z μ, ?_, ?_⟩\n · have hbound : ∀ n, eLpNorm (Z n) 1 μ ≤ (M.toNNReal : ℝ≥0∞) := by\n intro n\n rw [eLpNorm_one_eq_lintegral_enorm,\n ← ofReal_integral_norm_eq_lintegral_enorm (hZ.submartingale.integrable n)]\n have hle : (∫ x, ‖Z n x‖ ∂μ) ≤ M := by\n simpa [Real.norm_eq_abs] using hbdd n\n calc ENNReal.ofReal (∫ x, ‖Z n x‖ ∂μ)\n ≤ ENNReal.ofReal M := ENNReal.ofReal_le_ofReal hle\n _ = (M.toNNReal : ℝ≥0∞) := rfl\n exact memLp_one_iff_integrable.mp (hZ.submartingale.memLp_limitProcess hbound)\n · have hbound : ∀ n, eLpNorm (Z n) 1 μ ≤ (M.toNNReal : ℝ≥0∞) := by\n intro n\n rw [eLpNorm_one_eq_lintegral_enorm,\n ← ofReal_integral_norm_eq_lintegral_enorm (hZ.submartingale.integrable n)]\n have hle : (∫ x, ‖Z n x‖ ∂μ) ≤ M := by\n simpa [Real.norm_eq_abs] using hbdd n\n calc ENNReal.ofReal (∫ x, ‖Z n x‖ ∂μ)\n ≤ ENNReal.ofReal M := ENNReal.ofReal_le_ofReal hle\n _ = (M.toNNReal : ℝ≥0∞) := rfl\n exact hZ.submartingale.ae_tendsto_limitProcess hbound", "time_seconds": 49, "topic": "Martingales & stopping" }, { "question_id": 414, "problem_name": "stochastic_processes_supermartingale", "informal_statement": "Consider biased gambler’s ruin: at each step, the gambler gains one dollar\n with probability p and losses one dollar with probability (1 − p).\n Let X_n be the money in purse at time n.\n\n Show that if p < 1/2, then (X_n) is a supermartingale.", "lean_statement": "theorem q414_gambler_supermartingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hadap : Adapted ℱ X) (hint : ∀ n, Integrable (X n) μ) (p : ℝ) (hp : p < 1 / 2)\n (hstep : IsConstDrift μ ℱ X (2 * p - 1)) :\n Supermartingale X ℱ μ := by sorry", "true_to_source": "literal", "proof": "theorem q414_gambler_supermartingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (X : ℕ → Ω → ℝ) (hadap : Adapted ℱ X) (hint : ∀ n, Integrable (X n) μ) (p : ℝ) (hp : p < 1 / 2)\n (hstep : IsConstDrift μ ℱ X (2 * p - 1)) :\n Supermartingale X ℱ μ := by\n apply supermartingale_of_condExp_sub_nonneg_nat hadap.stronglyAdapted hint\n intro i\n have e : (X i - X (i + 1)) = -(fun ω => X (i + 1) ω - X i ω) := by\n funext ω; simp only [Pi.sub_apply, Pi.neg_apply]; ring\n have key : μ[X i - X (i + 1) | ℱ i] =ᵐ[μ] fun _ => (1 - 2 * p) := by\n rw [e]\n refine (condExp_neg (fun ω => X (i + 1) ω - X i ω) (ℱ i)).trans ?_\n refine ((hstep i).neg).trans ?_\n refine ae_of_all μ (fun ω => ?_)\n simp only [Pi.neg_apply]; ring\n refine Filter.EventuallyLE.trans_eq ?_ key.symm\n refine ae_of_all μ (fun ω => ?_)\n simp only [Pi.zero_apply]\n linarith", "time_seconds": 106, "topic": "Martingales & stopping" }, { "question_id": 415, "problem_name": "stochastic_processes_supermartingale_stopping_time_5", "informal_statement": "Let (X_n)_{n ≥ 0} be a supermartingale, and let T be a stopping time with 𝔼[T] < ∞.\n You are given that X has bounded increments, i.e. ∃M > 0 such that |X_{n+1} − X_n| ≤ M for all n.\n\n Show that 𝔼[X_T] ≤ 𝔼[X_0].", "lean_statement": "theorem q415_super_bdd_incr (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Supermartingale X ℱ μ) (T : Ω → ℕ)\n (hT : IsStoppingTime ℱ (natStop T)) (hET : Integrable (fun ω => (T ω : ℝ)) μ)\n (M : ℝ) (hincr : ∀ n, ∀ᵐ ω ∂μ, |X (n + 1) ω - X n ω| ≤ M) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ ≤ ∫ ω, X 0 ω ∂μ := by sorry", "true_to_source": "literal", "proof": "theorem q415_super_bdd_incr (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n [SigmaFiniteFiltration μ ℱ] (X : ℕ → Ω → ℝ) (hX : Supermartingale X ℱ μ) (T : Ω → ℕ)\n (hT : IsStoppingTime ℱ (natStop T)) (hET : Integrable (fun ω => (T ω : ℝ)) μ)\n (M : ℝ) (hincr : ∀ n, ∀ᵐ ω ∂μ, |X (n + 1) ω - X n ω| ≤ M) :\n ∫ ω, stoppedValue X (natStop T) ω ∂μ ≤ ∫ ω, X 0 ω ∂μ := by\n have hM : 0 ≤ M := by\n obtain ⟨ω, hω⟩ := (hincr 0).exists\n exact le_trans (abs_nonneg _) hω\n have htele : ∀ (k : ℕ) (ω : Ω), (∀ i ∈ Finset.range k, |X (i + 1) ω - X i ω| ≤ M) →\n |X k ω| ≤ |X 0 ω| + M * k := by\n intro k ω h\n induction k with\n | zero => simp\n | succ n ih =>\n have hn : ∀ i ∈ Finset.range n, |X (i + 1) ω - X i ω| ≤ M := fun i hi =>\n h i (Finset.mem_range.mpr (by simp at hi ⊢; omega))\n have hstep : |X (n + 1) ω - X n ω| ≤ M := h n (Finset.mem_range.mpr (Nat.lt_succ_self n))\n have hih := ih hn\n have htri : |X (n + 1) ω| ≤ |X n ω| + |X (n + 1) ω - X n ω| := by\n have := abs_sub_abs_le_abs_sub (X (n + 1) ω) (X n ω); linarith\n push_cast\n calc |X (n + 1) ω| ≤ |X n ω| + |X (n + 1) ω - X n ω| := htri\n _ ≤ (|X 0 ω| + M * n) + M := by gcongr\n _ = |X 0 ω| + M * (n + 1) := by ring\n set Y : Ω → ℝ := fun ω => |X 0 ω| + M * (T ω : ℝ) with hY\n have hYint : Integrable Y μ := by\n apply Integrable.add\n · exact (hX.integrable 0).abs\n · exact hET.const_mul M\n set F : ℕ → Ω → ℝ := fun N ω => X (min (T ω) N) ω with hF\n have hπ : ∀ N, IsStoppingTime ℱ (natStop (fun ω => min (T ω) N)) := by\n intro N; have h := hT.min_const N; convert h using 2\n have hbdd : ∀ N, ∀ ω, natStop (fun ω => min (T ω) N) ω ≤ (N : ℕ∞) := by\n intro N ω; simp only [Auto.natStop]; exact Nat.cast_le.mpr (min_le_right _ _)\n have hbound : ∀ N, ∫ ω, F N ω ∂μ ≤ ∫ ω, X 0 ω ∂μ := by\n intro N\n have hσ : IsStoppingTime ℱ (fun _ : Ω => ((0:ℕ) : ℕ∞)) := isStoppingTime_const ℱ 0\n have hle : (fun _ : Ω => ((0:ℕ) : ℕ∞)) ≤ natStop (fun ω => min (T ω) N) := by\n intro ω; simp [Auto.natStop]\n have h := hX.neg.expected_stoppedValue_mono hσ (hπ N) hle (hbdd N)\n have huntop : WithTop.untopA ((0:ℕ):ℕ∞) = 0 := rfl\n simp only [stoppedValue, Pi.neg_apply, integral_neg, huntop] at h\n have hrw : (fun a => X (natStop (fun ω => min (T ω) N) a).untopA a) = F N := by\n ext a; simp only [hF, Auto.natStop]; rfl\n rw [hrw] at h\n linarith [h]\n have hFint : ∀ N, Integrable (F N) μ := by\n intro N\n have := (hX.neg.integrable_stoppedValue (hπ N) (hbdd N)).neg\n rw [show (-stoppedValue (-X) (natStop (fun ω => min (T ω) N))) = F N by\n ext a; simp only [hF, stoppedValue, Pi.neg_apply, Auto.natStop, neg_neg]; rfl] at this\n exact this\n have h_bound : ∀ N, ∀ᵐ ω ∂μ, ‖F N ω‖ ≤ Y ω := by\n intro N\n have hall : ∀ᵐ ω ∂μ, ∀ i ∈ Finset.range N, |X (i + 1) ω - X i ω| ≤ M := by\n apply (ae_ball_iff (Finset.range N).countable_toSet).mpr\n intro i _; exact hincr i\n filter_upwards [hall] with ω hω\n have hk : ∀ i ∈ Finset.range (min (T ω) N), |X (i + 1) ω - X i ω| ≤ M := by\n intro i hi\n exact hω i (Finset.mem_range.mpr (lt_of_lt_of_le (Finset.mem_range.mp hi) (min_le_right _ _)))\n have hle1 := htele (min (T ω) N) ω hk\n have hle2 : ((min (T ω) N : ℕ) : ℝ) ≤ (T ω : ℝ) := by exact_mod_cast min_le_left _ _\n simp only [hF, Real.norm_eq_abs, hY]\n calc |X (min (T ω) N) ω| ≤ |X 0 ω| + M * ((min (T ω) N : ℕ) : ℝ) := hle1\n _ ≤ |X 0 ω| + M * (T ω : ℝ) := by gcongr\n have h_lim : ∀ᵐ ω ∂μ, Tendsto (fun N => F N ω) atTop (nhds (stoppedValue X (natStop T) ω)) := by\n filter_upwards with ω\n have hv : stoppedValue X (natStop T) ω = X (T ω) ω := by\n simp only [stoppedValue, Auto.natStop]; rfl\n rw [hv]\n apply tendsto_atTop_of_eventually_const (i₀ := T ω)\n intro N hN; simp only [hF]; rw [min_eq_left hN]\n have htend := tendsto_integral_of_dominated_convergence Y\n (fun N => (hFint N).aestronglyMeasurable) hYint h_bound h_lim\n exact le_of_tendsto' htend hbound", "time_seconds": 40, "topic": "Martingales & stopping" }, { "question_id": 420, "problem_name": "stochastic_processes_time_reversal_3", "informal_statement": "Let (X_n) be an irreducible Markov chain with transition matrix P and stationary distribution π.\n Define P' to be P'(x, y) = π(y)P(y, x) / π(x).\n\n Show that P' is stochastic.", "lean_statement": "theorem q420_time_reversal_stochastic {Ω : Type*} [Fintype Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (π : Ω → ℝ) (hπpos : ∀ x, 0 < π x)\n (hπ : IsStationary P π) (P' : Matrix Ω Ω ℝ) (hP' : IsReversal P π P') :\n IsStochastic P' := by sorry", "true_to_source": "literal", "proof": "theorem q420_time_reversal_stochastic {Ω : Type*} [Fintype Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (π : Ω → ℝ) (hπpos : ∀ x, 0 < π x)\n (hπ : IsStationary P π) (P' : Matrix Ω Ω ℝ) (hP' : IsReversal P π P') :\n IsStochastic P' := by\n constructor\n · intro x y\n rw [hP' x y]\n apply div_nonneg\n · exact mul_nonneg (le_of_lt (hπpos y)) (hP.1 y x)\n · exact le_of_lt (hπpos x)\n · intro x\n simp_rw [hP' x]\n rw [← Finset.sum_div, hπ.2.2 x]\n exact div_self (ne_of_gt (hπpos x))", "time_seconds": 60, "topic": "Markov chains (finite & countable)" }, { "question_id": 424, "problem_name": "stochastic_processes_tilted_sum_relation", "informal_statement": "Let (S_n) be a random walk with i.i.d. steps X, semi-invariant generating function γ(r), and tilted PMF q_{X,r}(x) = p_X(x) exp(rx − γ(r)).\n\nShow that the n-fold tilted sum satisfies q_{S_n,r}(s) = p_{S_n}(s) exp(rs − n γ(r)).", "lean_statement": "theorem q424_tilted_sum_relation\n {k : ℕ} (v : Fin k → ℝ) (p : Fin k → ℝ)\n (hp0 : ∀ x, 0 ≤ p x) (hpsum : ∑ x, p x = 1)\n (r : ℝ) (γ : ℝ) (hγ : γ = Real.log (∑ x, p x * Real.exp (r * v x)))\n (q : Fin k → ℝ) (hq : ∀ x, q x = p x * Real.exp (r * v x - γ))\n (n : ℕ) (s : ℝ) :\n (∑ w ∈ Finset.univ.filter (fun w : Fin n → Fin k => (∑ i, v (w i)) = s), ∏ i, q (w i))\n = (∑ w ∈ Finset.univ.filter (fun w : Fin n → Fin k => (∑ i, v (w i)) = s), ∏ i, p (w i))\n * Real.exp (r * s - (n : ℝ) * γ) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q424_tilted_sum_relation\n {k : ℕ} (v : Fin k → ℝ) (p : Fin k → ℝ)\n (hp0 : ∀ x, 0 ≤ p x) (hpsum : ∑ x, p x = 1)\n (r : ℝ) (γ : ℝ) (hγ : γ = Real.log (∑ x, p x * Real.exp (r * v x)))\n (q : Fin k → ℝ) (hq : ∀ x, q x = p x * Real.exp (r * v x - γ))\n (n : ℕ) (s : ℝ) :\n (∑ w ∈ Finset.univ.filter (fun w : Fin n → Fin k => (∑ i, v (w i)) = s), ∏ i, q (w i))\n = (∑ w ∈ Finset.univ.filter (fun w : Fin n → Fin k => (∑ i, v (w i)) = s), ∏ i, p (w i))\n * Real.exp (r * s - (n : ℝ) * γ) := by\n rw [Finset.sum_mul]\n apply Finset.sum_congr rfl\n intro w hw\n rw [Finset.mem_filter] at hw\n have hsum : (∑ i, v (w i)) = s := hw.2\n have e1 : (∏ i, q (w i)) = (∏ i, p (w i)) * ∏ i, Real.exp (r * v (w i) - γ) := by\n rw [← Finset.prod_mul_distrib]\n exact Finset.prod_congr rfl (fun i _ => hq (w i))\n have e2 : (∏ i, Real.exp (r * v (w i) - γ)) = Real.exp (r * s - (n : ℝ) * γ) := by\n rw [← Real.exp_sum]\n congr 1\n rw [Finset.sum_sub_distrib, ← Finset.mul_sum, Finset.sum_const, Finset.card_univ,\n Fintype.card_fin, nsmul_eq_mul, hsum]\n rw [e1, e2]", "time_seconds": 86, "topic": "Random walks & large deviations" }, { "question_id": 427, "problem_name": "stochastic_processes_mm1_expected_queue_number", "informal_statement": "Consider an M/M/1 queue in steady state with ρ = λ/μ < 1.\n\nShow that the expected number of customers waiting in queue (excluding the one in service) is ρ^2/(1 − ρ).", "lean_statement": "theorem q427_mm1_expected_queue_number\n (lam mu : ℝ) (hlam : 0 < lam) (hmu : 0 < mu)\n (ρ : ℝ) (hρ : ρ = lam / mu) (hρ0 : 0 < ρ) (hρ1 : ρ < 1)\n (p : ℕ → ℝ) (hp : ∀ i, p i = (1 - ρ) * ρ ^ i) :\n ∑' k : ℕ, (k : ℝ) * p (k + 1) = ρ ^ 2 / (1 - ρ) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q427_mm1_expected_queue_number\n (lam mu : ℝ) (hlam : 0 < lam) (hmu : 0 < mu)\n (ρ : ℝ) (hρ : ρ = lam / mu) (hρ0 : 0 < ρ) (hρ1 : ρ < 1)\n (p : ℕ → ℝ) (hp : ∀ i, p i = (1 - ρ) * ρ ^ i) :\n ∑' k : ℕ, (k : ℝ) * p (k + 1) = ρ ^ 2 / (1 - ρ) := by\n have hnorm : ‖ρ‖ < 1 := by rw [Real.norm_eq_abs, abs_of_pos hρ0]; exact hρ1\n have key : ∑' k : ℕ, (k:ℝ) * ρ^k = ρ / (1-ρ)^2 :=\n tsum_coe_mul_geometric_of_norm_lt_one hnorm\n have h1 : (1:ℝ) - ρ ≠ 0 := by linarith\n have hcongr : ∀ k : ℕ, (k:ℝ) * p (k+1) = (1-ρ)*ρ * ((k:ℝ) * ρ^k) := by\n intro k; rw [hp (k+1)]; ring\n rw [tsum_congr hcongr, tsum_mul_left, key]\n field_simp", "time_seconds": 172, "topic": "Continuous-time Markov & queues" }, { "question_id": 428, "problem_name": "stochastic_processes_detailed_balance", "informal_statement": "Show that any distribution π satisfying the detailed balance equations is stationary for P.", "lean_statement": "theorem q428_detailed_balance {Ω : Type*} [Fintype Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (π : Ω → ℝ)\n (hπnn : ∀ x, 0 ≤ π x) (hπ1 : ∑ x, π x = 1) (hdb : DetailedBalance P π) :\n IsStationary P π := by sorry", "true_to_source": "literal", "proof": "theorem q428_detailed_balance {Ω : Type*} [Fintype Ω]\n (P : Matrix Ω Ω ℝ) (hP : IsStochastic P) (π : Ω → ℝ)\n (hπnn : ∀ x, 0 ≤ π x) (hπ1 : ∑ x, π x = 1) (hdb : DetailedBalance P π) :\n IsStationary P π := by\n refine ⟨hπnn, hπ1, ?_⟩\n intro y\n calc ∑ x, π x * P x y = ∑ x, π y * P y x := Finset.sum_congr rfl (fun x _ => hdb x y)\n _ = π y * ∑ x, P y x := by rw [Finset.mul_sum]\n _ = π y * 1 := by rw [hP.2 y]\n _ = π y := by ring", "time_seconds": 54, "topic": "Markov chains (finite & countable)" }, { "question_id": 436, "problem_name": "stochastic_processes_convex_function_submartingale", "informal_statement": "Let (Z_n) be a martingale and let h be a convex function with E[|h(Z_n)|] < ∞ for all n.\n\nShow that (h(Z_n)) is a submartingale.", "lean_statement": "theorem q436_convex_submartingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (Z : ℕ → Ω → ℝ) (hZ : Martingale Z ℱ μ) (h : ℝ → ℝ) (hconv : ConvexOn ℝ Set.univ h)\n (hint : ∀ n, Integrable (fun ω => h (Z n ω)) μ) :\n Submartingale (fun n ω => h (Z n ω)) ℱ μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q436_convex_submartingale (μ : Measure Ω) [IsProbabilityMeasure μ] (ℱ : Filtration ℕ m0)\n (Z : ℕ → Ω → ℝ) (hZ : Martingale Z ℱ μ) (h : ℝ → ℝ) (hconv : ConvexOn ℝ Set.univ h)\n (hint : ∀ n, Integrable (fun ω => h (Z n ω)) μ) :\n Submartingale (fun n ω => h (Z n ω)) ℱ μ := by\n have hcont : Continuous h := continuousOn_univ.mp (hconv.continuousOn isOpen_univ)\n refine ⟨fun n => hcont.comp_stronglyMeasurable (hZ.1 n), fun i j hij => ?_, hint⟩\n have hjen := hconv.map_condExp_le_of_finiteDimensional (ℱ.le i) (hZ.integrable j) (hint j)\n have hmart := hZ.condExp_ae_eq hij\n filter_upwards [hjen, hmart] with ω hω hω2\n simp only [Function.comp_apply] at hω\n rw [hω2] at hω\n exact hω", "time_seconds": 173, "topic": "Martingales & stopping" }, { "question_id": 438, "problem_name": "stochastic_processes_stopped_supermartingale_bounds", "informal_statement": "Let (Z_n) be a supermartingale and J a stopping trial with stopped process (Z_n*).\n\nShow that E[Z_1] ≥ E[Z_n*] ≥ E[Z_n] for all n.", "lean_statement": "theorem q438_stopped_supermartingale_bounds (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ]\n (Z : ℕ → Ω → ℝ) (hZ : Supermartingale Z ℱ μ)\n (J : Ω → ℕ) (hJ : IsStoppingTime ℱ (natStop J)) (n : ℕ) :\n ∫ ω, Z n ω ∂μ ≤ ∫ ω, stoppedProcess Z (natStop J) n ω ∂μ ∧\n ∫ ω, stoppedProcess Z (natStop J) n ω ∂μ ≤ ∫ ω, Z 0 ω ∂μ := by\n sorry", "true_to_source": "literal", "proof": "theorem q438_stopped_supermartingale_bounds (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ]\n (Z : ℕ → Ω → ℝ) (hZ : Supermartingale Z ℱ μ)\n (J : Ω → ℕ) (hJ : IsStoppingTime ℱ (natStop J)) (n : ℕ) :\n ∫ ω, Z n ω ∂μ ≤ ∫ ω, stoppedProcess Z (natStop J) n ω ∂μ ∧\n ∫ ω, stoppedProcess Z (natStop J) n ω ∂μ ≤ ∫ ω, Z 0 ω ∂μ := by\n -- (b): stopped process is a supermartingale, so E[sp n] ≤ E[sp 0] = E[Z 0]\n have hsuper : Supermartingale (stoppedProcess Z (natStop J)) ℱ μ := by\n have h1 : Submartingale (stoppedProcess (-Z) (natStop J)) ℱ μ := hZ.neg.stoppedProcess hJ\n have h3 : -(stoppedProcess (-Z) (natStop J)) = stoppedProcess Z (natStop J) := by\n funext i ω; simp [MeasureTheory.stoppedProcess]\n have h2 := h1.neg\n rwa [h3] at h2\n have hsp0 : stoppedProcess Z (natStop J) 0 = Z 0 := by\n funext ω; simp [MeasureTheory.stoppedProcess]\n have hb := hsuper.setIntegral_le (Nat.zero_le n) (s := Set.univ) MeasurableSet.univ\n rw [setIntegral_univ, setIntegral_univ, hsp0] at hb\n -- (a): via optional stopping on submartingale -Z\n have hgsub : Submartingale (-Z) ℱ μ := hZ.neg\n have hσst : IsStoppingTime ℱ (fun ω => min ((n : ℕ∞)) (natStop J ω)) :=\n (isStoppingTime_const ℱ n).min hJ\n have hπst : IsStoppingTime ℱ (fun _ : Ω => (n : ℕ∞)) := isStoppingTime_const ℱ n\n have hle : (fun ω => min ((n : ℕ∞)) (natStop J ω)) ≤ (fun _ : Ω => (n : ℕ∞)) :=\n fun ω => min_le_left _ _\n have ha := hgsub.expected_stoppedValue_mono hσst hπst hle (N := n) (fun ω => le_refl _)\n have eσ : stoppedValue (-Z) (fun ω => min ((n : ℕ∞)) (natStop J ω))\n = -(stoppedProcess Z (natStop J) n) := by\n funext ω; rfl\n have eπ : stoppedValue (-Z) (fun _ : Ω => (n : ℕ∞)) = -(Z n) := by\n funext ω; rfl\n rw [eσ, eπ] at ha\n simp only [Pi.neg_apply, integral_neg] at ha\n constructor\n · linarith [ha]\n · exact hb", "time_seconds": 650, "topic": "Martingales & stopping" }, { "question_id": 440, "problem_name": "stochastic_processes_yule_classification", "informal_statement": "Consider the pure-birth (Yule) process with holding-time rate iλ in state i started from one individual, viewed as a countable-state Markov process.\n\nShow that the process is transient (every state is visited at most finitely often with probability 1).", "lean_statement": "theorem q440_yule_transient\n (P : ℕ → ℕ → ℝ) (hP0 : ∀ i j, 0 ≤ P i j)\n (hPup : ∀ i, P i (i + 1) = 1)\n (hPoff : ∀ i j, j ≠ i + 1 → P i j = 0)\n (p : ℕ → ℕ → ℕ → ℝ)\n (hp0 : ∀ n i j, 0 ≤ p n i j)\n (hpid : ∀ i j, p 0 i j = if i = j then 1 else 0)\n (hp1 : p 1 = P)\n (hCK : ∀ n i j, p (n + 1) i j = ∑' k, p n i k * p 1 k j) :\n ∀ x : ℕ, (∀ n, 1 ≤ n → p n x x = 0) ∧ Summable (fun n => p n x x) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q440_yule_transient\n (P : ℕ → ℕ → ℝ) (hP0 : ∀ i j, 0 ≤ P i j)\n (hPup : ∀ i, P i (i + 1) = 1)\n (hPoff : ∀ i j, j ≠ i + 1 → P i j = 0)\n (p : ℕ → ℕ → ℕ → ℝ)\n (hp0 : ∀ n i j, 0 ≤ p n i j)\n (hpid : ∀ i j, p 0 i j = if i = j then 1 else 0)\n (hp1 : p 1 = P)\n (hCK : ∀ n i j, p (n + 1) i j = ∑' k, p n i k * p 1 k j) :\n ∀ x : ℕ, (∀ n, 1 ≤ n → p n x x = 0) ∧ Summable (fun n => p n x x) := by\n have hform : ∀ n i j, p n i j = if j = i + n then (1 : ℝ) else 0 := by\n intro n\n induction n with\n | zero =>\n intro i j\n rw [hpid, Nat.add_zero]\n by_cases h : i = j\n · rw [if_pos h, if_pos h.symm]\n · rw [if_neg h, if_neg (fun hh => h hh.symm)]\n | succ m ih =>\n intro i j\n rw [hCK m i j, tsum_eq_single (i + m)]\n · rw [ih i (i + m), if_pos rfl, one_mul, hp1]\n by_cases hj : j = (i + m) + 1\n · rw [hj, hPup (i + m), if_pos (by omega)]\n · rw [hPoff (i + m) j hj, if_neg (by omega)]\n · intro k hk\n rw [ih i k, if_neg hk, zero_mul]\n intro x\n refine ⟨?_, ?_⟩\n · intro n hn\n rw [hform n x x, if_neg (by omega)]\n · apply summable_of_ne_finset_zero (s := {(0 : ℕ)})\n intro n hn\n simp only [Finset.mem_singleton] at hn\n rw [hform n x x, if_neg (by omega)]", "time_seconds": 123, "topic": "Continuous-time Markov & queues" }, { "question_id": 444, "problem_name": "stochastic_processes_mm1_process_stationary", "informal_statement": "Consider an M/M/1 queue as a continuous-time birth-death process with arrival rate λ, service rate μ, and ρ = λ/μ < 1.\n\nShow that the steady-state distribution is p_i = (1 − ρ) ρ^i and that the expected number in the system is E[X(t)] = ρ/(1 − ρ).", "lean_statement": "theorem q444_mm1_stationary\n (lam mu : ℝ) (hlam : 0 < lam) (hmu : 0 < mu)\n (ρ : ℝ) (hρ : ρ = lam / mu) (hρ1 : ρ < 1)\n (p : ℕ → ℝ) (hp0 : ∀ i, 0 ≤ p i) (hpsum : ∑' i, p i = 1)\n (hbal0 : p 0 * lam = p 1 * mu)\n (hbal : ∀ n, p (n + 1) * (lam + mu) = p n * lam + p (n + 2) * mu) :\n (∀ i, p i = (1 - ρ) * ρ ^ i) ∧ (∑' i : ℕ, (i : ℝ) * p i = ρ / (1 - ρ)) := by\n sorry", "true_to_source": "abstract", "proof": "theorem q444_mm1_stationary\n (lam mu : ℝ) (hlam : 0 < lam) (hmu : 0 < mu)\n (ρ : ℝ) (hρ : ρ = lam / mu) (hρ1 : ρ < 1)\n (p : ℕ → ℝ) (hp0 : ∀ i, 0 ≤ p i) (hpsum : ∑' i, p i = 1)\n (hbal0 : p 0 * lam = p 1 * mu)\n (hbal : ∀ n, p (n + 1) * (lam + mu) = p n * lam + p (n + 2) * mu) :\n (∀ i, p i = (1 - ρ) * ρ ^ i) ∧ (∑' i : ℕ, (i : ℝ) * p i = ρ / (1 - ρ)) := by\n have hmune : mu ≠ 0 := ne_of_gt hmu\n have hρpos : 0 < ρ := by rw [hρ]; exact div_pos hlam hmu\n have hρnonneg : 0 ≤ ρ := le_of_lt hρpos\n have h1mρ : (1 : ℝ) - ρ ≠ 0 := by\n have : 0 < 1 - ρ := by linarith\n exact ne_of_gt this\n -- local balance\n have hlocal : ∀ n, p (n + 1) * mu = p n * lam := by\n intro n\n induction n with\n | zero => linarith [hbal0]\n | succ k ih =>\n have hb := hbal k\n rw [mul_add] at hb\n linarith [ih, hb]\n -- geometric form\n have hgeo : ∀ n, p n = ρ ^ n * p 0 := by\n intro n\n induction n with\n | zero => simp\n | succ k ih =>\n have hstep : p (k + 1) = p k * ρ := by\n have h := hlocal k\n rw [hρ]\n field_simp\n linarith [h]\n rw [hstep, ih, pow_succ]\n ring\n -- p 0 = 1 - ρ\n have hcompute : ∑' i, p i = (1 - ρ)⁻¹ * p 0 := by\n rw [tsum_congr hgeo, tsum_mul_right, tsum_geometric_of_lt_one hρnonneg hρ1]\n have hsum2 : (1 - ρ)⁻¹ * p 0 = 1 := hcompute ▸ hpsum\n have hp0val : p 0 = 1 - ρ := by\n field_simp at hsum2\n linarith [hsum2]\n have hpart1 : ∀ i, p i = (1 - ρ) * ρ ^ i := by\n intro i\n rw [hgeo i, hp0val]; ring\n refine ⟨hpart1, ?_⟩\n have hnorm : ‖ρ‖ < 1 := by\n rw [Real.norm_eq_abs, abs_of_nonneg hρnonneg]; exact hρ1\n calc ∑' i : ℕ, (i : ℝ) * p i\n = ∑' i : ℕ, (1 - ρ) * ((i : ℝ) * ρ ^ i) := by\n apply tsum_congr; intro i; rw [hpart1 i]; ring\n _ = (1 - ρ) * ∑' i : ℕ, (i : ℝ) * ρ ^ i := tsum_mul_left\n _ = (1 - ρ) * (ρ / (1 - ρ) ^ 2) := by\n rw [tsum_coe_mul_geometric_of_norm_lt_one hnorm]\n _ = ρ / (1 - ρ) := by field_simp", "time_seconds": 239, "topic": "Continuous-time Markov & queues" }, { "question_id": 447, "problem_name": "stochastic_processes_ito_integral_linearity_simple", "informal_statement": "Let X and Y be simple adapted processes in L^2 and let α,β be constants.\n\nShow that the Itô integral of simple processes is linear: I_t(αX + βY) = α I_t(X) + β I_t(Y).", "lean_statement": "theorem q447_ito_simple_linear (μ : Measure Ω)\n (B : ℝ → Ω → ℝ) (s : ℕ → ℝ) (N : ℕ)\n (ξ η : ℕ → Ω → ℝ) (a b : ℝ)\n (I : (ℕ → Ω → ℝ) → Ω → ℝ)\n (hI : ∀ (c : ℕ → Ω → ℝ) ω,\n I c ω = ∑ j ∈ Finset.range N, c j ω * (B (s (j + 1)) ω - B (s j) ω)) :\n ∀ ω, I (fun j ω => a * ξ j ω + b * η j ω) ω = a * I ξ ω + b * I η ω := by\n sorry", "true_to_source": "abstract", "proof": "theorem q447_ito_simple_linear (μ : Measure Ω)\n (B : ℝ → Ω → ℝ) (s : ℕ → ℝ) (N : ℕ)\n (ξ η : ℕ → Ω → ℝ) (a b : ℝ)\n (I : (ℕ → Ω → ℝ) → Ω → ℝ)\n (hI : ∀ (c : ℕ → Ω → ℝ) ω,\n I c ω = ∑ j ∈ Finset.range N, c j ω * (B (s (j + 1)) ω - B (s j) ω)) :\n ∀ ω, I (fun j ω => a * ξ j ω + b * η j ω) ω = a * I ξ ω + b * I η ω := by\n intro ω\n rw [hI, hI, hI, Finset.mul_sum, Finset.mul_sum, ← Finset.sum_add_distrib]\n apply Finset.sum_congr rfl\n intro j _\n ring", "time_seconds": 45, "topic": "Brownian motion & stochastic calculus" }, { "question_id": 90, "problem_name": "stochastic_processes_ito_integral_martingale_simple", "informal_statement": "Let X be a simple adapted process in L^2.\n\nShow that I_t(X) is a continuous square-integrable martingale, i.e. I_t(X) ∈ M^{2,c}.", "lean_statement": "theorem q90_ito_simple_is_M2c (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℝ m0) (B : ℝ → Ω → ℝ)\n (hBmart : Martingale B ℱ μ)\n (hBcont : ∀ ω, Continuous (fun t => B t ω))\n (hBL2 : ∀ t, MemLp (B t) 2 μ)\n (s : ℕ → ℝ) (hsmono : Monotone s) (N : ℕ)\n (ξ : ℕ → Ω → ℝ)\n (hξadap : ∀ j, StronglyMeasurable[ℱ (s j)] (ξ j))\n (hξbdd : ∀ j, ∃ C : ℝ, ∀ ω, |ξ j ω| ≤ C)\n (J : ℝ → Ω → ℝ)\n (hJ : ∀ t ω, J t ω =\n ∑ j ∈ Finset.range N, ξ j ω * (B (min t (s (j + 1))) ω - B (min t (s j)) ω)) :\n Martingale J ℱ μ ∧ (∀ ω, Continuous (fun t => J t ω)) ∧ (∀ t, MemLp (J t) 2 μ) := by\n sorry", "true_to_source": "abstract", "proof": "private lemma w90_condExp_ae_zero {Ω : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}\n (m : MeasurableSpace Ω) {f : Ω → ℝ} (hf : f =ᵐ[μ] 0) : μ[f | m] =ᵐ[μ] 0 := by\n have h := condExp_congr_ae (m := m) hf\n rwa [condExp_zero] at h\n\nprivate lemma w90_stop {Ω : Type*} {m0 : MeasurableSpace Ω} (μ : Measure Ω)\n [IsProbabilityMeasure μ] (ℱ : Filtration ℝ m0) (B : ℝ → Ω → ℝ)\n (hBmart : Martingale B ℱ μ) (hBL2 : ∀ t, MemLp (B t) 2 μ)\n (c i t : ℝ) (hit : i ≤ t) :\n μ[B (min t c) | (ℱ i : MeasurableSpace Ω)] =ᵐ[μ] B (min i c) := by\n rcases le_or_gt c i with hci | hci\n · have hSM : StronglyMeasurable[(ℱ i : MeasurableSpace Ω)] (B c) :=\n (hBmart.1 c).mono (ℱ.mono hci)\n have hEq : μ[B c | (ℱ i : MeasurableSpace Ω)] = B c :=\n condExp_of_stronglyMeasurable (ℱ.le i) hSM ((hBL2 c).integrable (by norm_num))\n rw [min_eq_right (hci.trans hit), min_eq_right hci]\n exact Filter.Eventually.of_forall (fun ω => congrFun hEq ω)\n · rw [min_eq_left hci.le]\n exact hBmart.condExp_ae_eq (le_min hit hci.le)\n\nprivate lemma w90_term {Ω : Type*} {m0 : MeasurableSpace Ω} (μ : Measure Ω)\n [IsProbabilityMeasure μ] (ℱ : Filtration ℝ m0) (B : ℝ → Ω → ℝ)\n (hBmart : Martingale B ℱ μ) (hBL2 : ∀ t, MemLp (B t) 2 μ)\n (a b : ℝ) (hab : a ≤ b) (g : Ω → ℝ) (C : ℝ)\n (hg_adap : StronglyMeasurable[(ℱ a : MeasurableSpace Ω)] g) (hg_bdd : ∀ ω, |g ω| ≤ C) :\n Martingale (fun t ω => g ω * (B (min t b) ω - B (min t a) ω)) ℱ μ := by\n have hBint : ∀ u, Integrable (B u) μ := fun u => (hBL2 u).integrable (by norm_num)\n have hDint : ∀ t, Integrable (fun ω => B (min t b) ω - B (min t a) ω) μ :=\n fun t => (hBint (min t b)).sub (hBint (min t a))\n have hgae : AEStronglyMeasurable g μ := (hg_adap.mono (ℱ.le a)).aestronglyMeasurable\n have hgbound : ∀ᵐ ω ∂μ, ‖g ω‖ ≤ C :=\n Filter.Eventually.of_forall (fun ω => by rw [Real.norm_eq_abs]; exact hg_bdd ω)\n have hMint : ∀ t, Integrable (fun ω => g ω * (B (min t b) ω - B (min t a) ω)) μ :=\n fun t => (hDint t).bdd_mul hgae hgbound\n refine ⟨?_, ?_⟩\n · intro t\n rcases le_or_gt a t with hat | hta\n · have hgt : StronglyMeasurable[(ℱ t : MeasurableSpace Ω)] g := hg_adap.mono (ℱ.mono hat)\n have hBb : StronglyMeasurable[(ℱ t : MeasurableSpace Ω)] (B (min t b)) :=\n (hBmart.1 (min t b)).mono (ℱ.mono (min_le_left t b))\n have hBa : StronglyMeasurable[(ℱ t : MeasurableSpace Ω)] (B (min t a)) :=\n (hBmart.1 (min t a)).mono (ℱ.mono (min_le_left t a))\n exact hgt.mul (hBb.sub hBa)\n · have heq : (fun ω => g ω * (B (min t b) ω - B (min t a) ω)) = (fun _ => (0 : ℝ)) := by\n funext ω\n rw [min_eq_left (le_of_lt (lt_of_lt_of_le hta hab)), min_eq_left (le_of_lt hta)]\n ring\n simp only [heq]\n exact stronglyMeasurable_const\n · intro i j hij\n rcases le_or_gt a i with hai | hia\n · have hgi : StronglyMeasurable[(ℱ i : MeasurableSpace Ω)] g := hg_adap.mono (ℱ.mono hai)\n have hpull := condExp_mul_of_stronglyMeasurable_left (f := g)\n (g := fun ω => B (min j b) ω - B (min j a) ω) hgi (hMint j) (hDint j)\n have hcs : μ[(fun ω => B (min j b) ω - B (min j a) ω) | (ℱ i : MeasurableSpace Ω)]\n =ᵐ[μ] fun ω => B (min i b) ω - B (min i a) ω := by\n have hsub := condExp_sub (hBint (min j b)) (hBint (min j a)) (ℱ i : MeasurableSpace Ω)\n have hb := w90_stop μ ℱ B hBmart hBL2 b i j hij\n have ha := w90_stop μ ℱ B hBmart hBL2 a i j hij\n exact hsub.trans (hb.sub ha)\n refine hpull.trans ?_\n exact EventuallyEq.mul EventuallyEq.rfl hcs\n · have hMi0 : (fun ω => g ω * (B (min i b) ω - B (min i a) ω)) =ᵐ[μ] 0 := by\n have heq : (fun ω => g ω * (B (min i b) ω - B (min i a) ω)) = (fun _ => (0 : ℝ)) := by\n funext ω\n rw [min_eq_left (le_of_lt (lt_of_lt_of_le hia hab)), min_eq_left (le_of_lt hia)]\n ring\n rw [heq]; exact EventuallyEq.rfl\n rcases le_or_gt a j with haj | hja\n · have hinner :\n μ[(fun ω => g ω * (B (min j b) ω - B (min j a) ω)) | (ℱ a : MeasurableSpace Ω)]\n =ᵐ[μ] 0 := by\n have hpull := condExp_mul_of_stronglyMeasurable_left (f := g)\n (g := fun ω => B (min j b) ω - B (min j a) ω) hg_adap (hMint j) (hDint j)\n have hcs : μ[(fun ω => B (min j b) ω - B (min j a) ω) | (ℱ a : MeasurableSpace Ω)]\n =ᵐ[μ] 0 := by\n have hsub := condExp_sub (hBint (min j b)) (hBint (min j a)) (ℱ a : MeasurableSpace Ω)\n have hb := w90_stop μ ℱ B hBmart hBL2 b a j haj\n have ha := w90_stop μ ℱ B hBmart hBL2 a a j haj\n refine hsub.trans ((hb.sub ha).trans ?_)\n refine Filter.Eventually.of_forall (fun ω => ?_)\n simp only [Pi.sub_apply, Pi.zero_apply, min_eq_left hab, min_self, sub_self]\n refine hpull.trans ?_\n refine (EventuallyEq.mul EventuallyEq.rfl hcs).trans ?_\n exact Filter.Eventually.of_forall (fun ω => by simp)\n refine (Filter.EventuallyEq.trans ?_ hMi0.symm)\n refine (Filtration.condExp_condExp\n (fun ω => g ω * (B (min j b) ω - B (min j a) ω)) ℱ hia.le).symm.trans ?_\n exact w90_condExp_ae_zero _ hinner\n · have hMj0 : (fun ω => g ω * (B (min j b) ω - B (min j a) ω)) =ᵐ[μ] 0 := by\n have heq : (fun ω => g ω * (B (min j b) ω - B (min j a) ω)) = (fun _ => (0 : ℝ)) := by\n funext ω\n rw [min_eq_left (le_of_lt (lt_of_lt_of_le hja hab)), min_eq_left (le_of_lt hja)]\n ring\n rw [heq]; exact EventuallyEq.rfl\n exact (w90_condExp_ae_zero _ hMj0).trans hMi0.symm\n\nprivate lemma w90_mart_sum {Ω : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}\n {ℱ : Filtration ℝ m0} (M : ℕ → ℝ → Ω → ℝ) (s : Finset ℕ)\n (h : ∀ j ∈ s, Martingale (M j) ℱ μ) :\n Martingale (fun t => ∑ j ∈ s, M j t) ℱ μ := by\n classical\n induction s using Finset.induction_on with\n | empty =>\n have he : (fun t => ∑ j ∈ (∅ : Finset ℕ), M j t) = (0 : ℝ → Ω → ℝ) := by\n funext t; simp\n rw [he]; exact martingale_zero ℝ ℱ μ\n | @insert j s' hjs ih =>\n have hj : Martingale (M j) ℱ μ := h j (Finset.mem_insert_self j s')\n have ih' : Martingale (fun t => ∑ k ∈ s', M k t) ℱ μ :=\n ih (fun k hk => h k (Finset.mem_insert_of_mem hk))\n have heq : (fun t => ∑ k ∈ insert j s', M k t) = (M j) + (fun t => ∑ k ∈ s', M k t) := by\n funext t; simp only [Finset.sum_insert hjs, Pi.add_apply]\n rw [heq]; exact hj.add ih'\n\ntheorem q90_ito_simple_is_M2c (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℝ m0) (B : ℝ → Ω → ℝ)\n (hBmart : Martingale B ℱ μ)\n (hBcont : ∀ ω, Continuous (fun t => B t ω))\n (hBL2 : ∀ t, MemLp (B t) 2 μ)\n (s : ℕ → ℝ) (hsmono : Monotone s) (N : ℕ)\n (ξ : ℕ → Ω → ℝ)\n (hξadap : ∀ j, StronglyMeasurable[ℱ (s j)] (ξ j))\n (hξbdd : ∀ j, ∃ C : ℝ, ∀ ω, |ξ j ω| ≤ C)\n (J : ℝ → Ω → ℝ)\n (hJ : ∀ t ω, J t ω =\n ∑ j ∈ Finset.range N, ξ j ω * (B (min t (s (j + 1))) ω - B (min t (s j)) ω)) :\n Martingale J ℱ μ ∧ (∀ ω, Continuous (fun t => J t ω)) ∧ (∀ t, MemLp (J t) 2 μ) := by\n refine ⟨?_, ?_, ?_⟩\n · have hJfun : J = fun t => ∑ j ∈ Finset.range N,\n (fun (t : ℝ) (ω : Ω) => ξ j ω * (B (min t (s (j + 1))) ω - B (min t (s j)) ω)) t := by\n funext t ω\n rw [hJ t ω, Finset.sum_apply]\n rw [hJfun]\n refine w90_mart_sum _ (Finset.range N) (fun j _ => ?_)\n obtain ⟨C, hC⟩ := hξbdd j\n exact w90_term μ ℱ B hBmart hBL2 (s j) (s (j + 1)) (hsmono (Nat.le_succ j)) (ξ j) C\n (hξadap j) hC\n · intro ω\n have hcontfun : (fun t => J t ω) = fun t => ∑ j ∈ Finset.range N,\n ξ j ω * (B (min t (s (j + 1))) ω - B (min t (s j)) ω) := by\n funext t; exact hJ t ω\n rw [hcontfun]\n apply continuous_finsetSum\n intro j _\n exact continuous_const.mul\n (((hBcont ω).comp (continuous_id.min continuous_const)).sub\n ((hBcont ω).comp (continuous_id.min continuous_const)))\n · intro t\n have hJt : J t = fun ω => ∑ j ∈ Finset.range N,\n ξ j ω * (B (min t (s (j + 1))) ω - B (min t (s j)) ω) := by\n funext ω; exact hJ t ω\n rw [hJt]\n apply memLp_finsetSum\n intro j _\n obtain ⟨C, hC⟩ := hξbdd j\n have hD : MemLp (fun ω => B (min t (s (j + 1))) ω - B (min t (s j)) ω) 2 μ :=\n (hBL2 (min t (s (j + 1)))).sub (hBL2 (min t (s j)))\n refine MemLp.of_le_mul (c := C) hD ?_ ?_\n · exact ((hξadap j).mono (ℱ.le (s j))).aestronglyMeasurable.mul hD.aestronglyMeasurable\n · exact Filter.Eventually.of_forall (fun ω => by\n simp only [Real.norm_eq_abs, abs_mul]\n exact mul_le_mul_of_nonneg_right (hC ω) (abs_nonneg _))", "time_seconds": 1953, "topic": "Brownian motion & stochastic calculus" }, { "question_id": 197, "problem_name": "stochastic_processes_renewal_reward", "informal_statement": "Let (N(t)) be a renewal process with finite mean interrenewal time X̄, and let R(t) ≥ 0 with almost surely locally integrable paths be a renewal-reward function for which the reward R_n earned in interrenewal interval n satisfies E[R_n] < ∞ and depends only on the nth interrenewal interval and the age within it.\n\nShow that lim_{t→∞} (1/t) ∫_0^t R(τ) dτ = E[R_n]/X̄ with probability 1.", "lean_statement": "theorem q197_renewal_reward (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hpos : ∀ i ω, 0 < X i ω)\n (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (hXint : Integrable (X 0) μ)\n (Xbar : ℝ) (hmean : ∫ ω, X 0 ω ∂μ = Xbar) (hXpos : 0 < Xbar)\n (S : ℕ → Ω → ℝ) (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, X i ω)\n (R : ℝ → Ω → ℝ)\n (hRlocal : ∀ᵐ ω ∂μ, ∀ T : ℝ, 0 ≤ T → IntegrableOn (fun t => R t ω) (Set.Ioc 0 T))\n (hRnn : ∀ τ ω, 0 ≤ R τ ω)\n (Rcyc : ℕ → Ω → ℝ) (hlink : ∀ n ω, Rcyc n ω = ∫ τ in (S n ω)..(S (n + 1) ω), R τ ω)\n (hRint : ∀ n, Integrable (Rcyc n) μ)\n (hRcycindep : iIndepFun Rcyc μ) (hRcycident : ∀ n, IdentDistrib (Rcyc n) (Rcyc 0) μ μ)\n (r : ℝ) (hr : ∀ n, ∫ ω, Rcyc n ω ∂μ = r) :\n ∀ᵐ ω ∂μ, Tendsto (fun t : ℝ => (1 / t) * ∫ τ in (0:ℝ)..t, R τ ω) atTop (𝓝 (r / Xbar)) := by\n sorry", "true_to_source": "abstract", "proof": "private lemma renewal_squeeze\n (g : ℝ → ℝ) (s : ℕ → ℝ) (L c : ℝ)\n (hc : 0 < c)\n (hs0 : s 0 = 0)\n (hs_mono : StrictMono s)\n (hs_ratio : Tendsto (fun n => s n / (n : ℝ)) atTop (𝓝 c))\n (hg_mono : ∀ a b : ℝ, 0 ≤ a → a ≤ b → g a ≤ g b)\n (hg_nonneg : ∀ t : ℝ, 0 ≤ t → 0 ≤ g t)\n (hratio : Tendsto (fun n => g (s n) / s n) atTop (𝓝 L)) :\n Tendsto (fun t => g t / t) atTop (𝓝 L) := by\n classical\n have hspos : ∀ n : ℕ, 1 ≤ n → 0 < s n := by\n intro n hn\n have := hs_mono (show 0 < n by omega)\n rwa [hs0] at this\n have hs_top : Tendsto s atTop atTop := by\n have hmul : Tendsto (fun n : ℕ => (s n / (n : ℝ)) * (n : ℝ)) atTop atTop :=\n Filter.Tendsto.pos_mul_atTop hc hs_ratio tendsto_natCast_atTop_atTop\n refine hmul.congr' ?_\n filter_upwards [eventually_ge_atTop 1] with n hn\n have hn0 : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (by omega)\n field_simp\n have hgrow : Tendsto (fun n => s (n + 1) / s n) atTop (𝓝 1) := by\n have h1 : Tendsto (fun n : ℕ => s (n + 1) / (↑(n + 1) : ℝ)) atTop (𝓝 c) :=\n hs_ratio.comp (Filter.tendsto_add_atTop_nat 1)\n have h2 : Tendsto (fun n : ℕ => (↑(n + 1) : ℝ) / (n : ℝ)) atTop (𝓝 1) := by\n have hz : Tendsto (fun n : ℕ => 1 + (n : ℝ)⁻¹) atTop (𝓝 (1 + 0)) :=\n tendsto_const_nhds.add tendsto_inv_atTop_nhds_zero_nat\n rw [add_zero] at hz\n refine hz.congr' ?_\n filter_upwards [eventually_ge_atTop 1] with n hn\n have hn0 : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (by omega)\n push_cast\n field_simp\n have h3 : Tendsto (fun n : ℕ =>\n (s (n + 1) / (↑(n + 1) : ℝ)) * ((↑(n + 1) : ℝ) / (n : ℝ)) / (s n / (n : ℝ)))\n atTop (𝓝 (c * 1 / c)) := (h1.mul h2).div hs_ratio (ne_of_gt hc)\n have hcc : c * 1 / c = 1 := by field_simp\n rw [hcc] at h3\n refine h3.congr' ?_\n filter_upwards [eventually_ge_atTop 1] with n hn\n have hn0 : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (by omega)\n have hn1 : (↑(n + 1) : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (by omega)\n have hsn : s n ≠ 0 := ne_of_gt (hspos n hn)\n field_simp\n have hex : ∀ t : ℝ, ∃ m : ℕ, t < s m := by\n intro t\n exact (hs_top.eventually (eventually_gt_atTop t)).exists\n set M : ℝ → ℕ := fun t => Nat.find (hex t) with hMdef\n have hMspec : ∀ t : ℝ, t < s (M t) := fun t => Nat.find_spec (hex t)\n have hMmin : ∀ (t : ℝ) (m : ℕ), m < M t → s m ≤ t := by\n intro t m hm\n have := Nat.find_min (hex t) hm\n push_neg at this\n exact this\n have hMpos : ∀ t : ℝ, 0 ≤ t → 1 ≤ M t := by\n intro t ht\n rcases Nat.eq_zero_or_pos (M t) with h | h\n · exfalso\n have hs := hMspec t\n rw [h, hs0] at hs\n linarith\n · exact h\n set N : ℝ → ℕ := fun t => M t - 1 with hNdef\n have hNeq : ∀ t : ℝ, 0 ≤ t → M t = N t + 1 := by\n intro t ht\n have := hMpos t ht\n simp only [hNdef]; omega\n have hNle : ∀ t : ℝ, 0 ≤ t → s (N t) ≤ t := by\n intro t ht\n apply hMmin\n have := hMpos t ht\n simp only [hNdef]; omega\n have hNlt : ∀ t : ℝ, 0 ≤ t → t < s (N t + 1) := by\n intro t ht\n have h := hMspec t\n rw [hNeq t ht] at h\n exact h\n have hN_top : Tendsto N atTop atTop := by\n rw [tendsto_atTop]\n intro K\n filter_upwards [eventually_ge_atTop (s K), eventually_ge_atTop (0 : ℝ)] with t ht ht0\n by_contra h\n push_neg at h\n have h1 : N t + 1 ≤ K := by omega\n have h2 : s (N t + 1) ≤ s K := hs_mono.monotone h1\n have h3 := hNlt t ht0\n linarith\n have hN1_top : Tendsto (fun t => N t + 1) atTop atTop :=\n tendsto_atTop_mono (fun t => Nat.le_succ (N t)) hN_top\n have hA : Tendsto (fun t => g (s (N t)) / s (N t)) atTop (𝓝 L) := by\n simpa [Function.comp] using hratio.comp hN_top\n have hC : Tendsto (fun t => g (s (N t + 1)) / s (N t + 1)) atTop (𝓝 L) := by\n simpa [Function.comp] using hratio.comp hN1_top\n have hDgrow : Tendsto (fun t => s (N t + 1) / s (N t)) atTop (𝓝 1) := by\n simpa [Function.comp] using hgrow.comp hN_top\n have hB : Tendsto (fun t => s (N t) / s (N t + 1)) atTop (𝓝 1) := by\n have : (fun t => s (N t) / s (N t + 1)) = (fun t => (s (N t + 1) / s (N t))⁻¹) := by\n funext t; rw [inv_div]\n rw [this]\n simpa using hDgrow.inv₀ (one_ne_zero)\n have hLo : Tendsto (fun t => g (s (N t)) / s (N t + 1)) atTop (𝓝 L) := by\n have hprod : Tendsto (fun t => (g (s (N t)) / s (N t)) * (s (N t) / s (N t + 1)))\n atTop (𝓝 L) := by simpa using hA.mul hB\n refine hprod.congr' ?_\n filter_upwards [hN_top.eventually_ge_atTop 1] with t ht\n have hsn : (s (N t)) ≠ 0 := ne_of_gt (hspos (N t) ht)\n have hsn1 : (s (N t + 1)) ≠ 0 := ne_of_gt (hspos (N t + 1) (by omega))\n field_simp\n have hHi : Tendsto (fun t => g (s (N t + 1)) / s (N t)) atTop (𝓝 L) := by\n have hprod : Tendsto (fun t => (g (s (N t + 1)) / s (N t + 1)) * (s (N t + 1) / s (N t)))\n atTop (𝓝 L) := by simpa using hC.mul hDgrow\n refine hprod.congr' ?_\n filter_upwards [hN_top.eventually_ge_atTop 1] with t ht\n have hsn : (s (N t)) ≠ 0 := ne_of_gt (hspos (N t) ht)\n have hsn1 : (s (N t + 1)) ≠ 0 := ne_of_gt (hspos (N t + 1) (by omega))\n field_simp\n apply tendsto_of_tendsto_of_tendsto_of_le_of_le' hLo hHi\n · filter_upwards [hN_top.eventually_ge_atTop 1, eventually_ge_atTop (0 : ℝ)] with t ht ht0\n have hsn : 0 < s (N t) := hspos (N t) ht\n have hle : s (N t) ≤ t := hNle t ht0\n have hlt : t < s (N t + 1) := hNlt t ht0\n have htpos : 0 < t := lt_of_lt_of_le hsn hle\n have hs1pos : 0 < s (N t + 1) := hspos (N t + 1) (by omega)\n rw [div_le_div_iff₀ hs1pos htpos]\n exact mul_le_mul (hg_mono _ _ (le_of_lt hsn) hle) (le_of_lt hlt) (le_of_lt htpos)\n (hg_nonneg _ (le_of_lt htpos))\n · filter_upwards [hN_top.eventually_ge_atTop 1, eventually_ge_atTop (0 : ℝ)] with t ht ht0\n have hsn : 0 < s (N t) := hspos (N t) ht\n have hle : s (N t) ≤ t := hNle t ht0\n have hlt : t < s (N t + 1) := hNlt t ht0\n have htpos : 0 < t := lt_of_lt_of_le hsn hle\n have hs1pos : 0 < s (N t + 1) := hspos (N t + 1) (by omega)\n rw [div_le_div_iff₀ htpos hsn]\n exact mul_le_mul (hg_mono _ _ (le_of_lt htpos) (le_of_lt hlt)) hle (le_of_lt hsn)\n (hg_nonneg _ (le_of_lt hs1pos))\n\ntheorem q197_renewal_reward (μ : Measure Ω) [IsProbabilityMeasure μ]\n (X : ℕ → Ω → ℝ) (hpos : ∀ i ω, 0 < X i ω)\n (hindep : iIndepFun X μ) (hident : ∀ i, IdentDistrib (X i) (X 0) μ μ)\n (hXint : Integrable (X 0) μ)\n (Xbar : ℝ) (hmean : ∫ ω, X 0 ω ∂μ = Xbar) (hXpos : 0 < Xbar)\n (S : ℕ → Ω → ℝ) (hS : ∀ n ω, S n ω = ∑ i ∈ Finset.range n, X i ω)\n (R : ℝ → Ω → ℝ)\n (hRlocal : ∀ᵐ ω ∂μ, ∀ T : ℝ, 0 ≤ T → IntegrableOn (fun t => R t ω) (Set.Ioc 0 T))\n (hRnn : ∀ τ ω, 0 ≤ R τ ω)\n (Rcyc : ℕ → Ω → ℝ) (hlink : ∀ n ω, Rcyc n ω = ∫ τ in (S n ω)..(S (n + 1) ω), R τ ω)\n (hRint : ∀ n, Integrable (Rcyc n) μ)\n (hRcycindep : iIndepFun Rcyc μ) (hRcycident : ∀ n, IdentDistrib (Rcyc n) (Rcyc 0) μ μ)\n (r : ℝ) (hr : ∀ n, ∫ ω, Rcyc n ω ∂μ = r) :\n ∀ᵐ ω ∂μ, Tendsto (fun t : ℝ => (1 / t) * ∫ τ in (0:ℝ)..t, R τ ω) atTop (𝓝 (r / Xbar)) := by\n have sllnX : ∀ᵐ ω ∂μ,\n Tendsto (fun n => (∑ i ∈ Finset.range n, X i ω) / (n : ℝ)) atTop (𝓝 Xbar) := by\n have h := strong_law_ae_real X hXint (fun i j hij => hindep.indepFun hij) hident\n rw [hmean] at h\n exact h\n have sllnR : ∀ᵐ ω ∂μ,\n Tendsto (fun n => (∑ i ∈ Finset.range n, Rcyc i ω) / (n : ℝ)) atTop (𝓝 r) := by\n have h := strong_law_ae_real Rcyc (hRint 0) (fun i j hij => hRcycindep.indepFun hij) hRcycident\n rw [hr 0] at h\n exact h\n filter_upwards [sllnX, sllnR, hRlocal] with ω hSX hSR hRloc\n have hII : ∀ a b : ℝ, 0 ≤ a → a ≤ b →\n IntervalIntegrable (fun τ => R τ ω) volume a b := by\n intro a b ha hab\n have hb : 0 ≤ b := le_trans ha hab\n rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hab]\n exact (hRloc b hb).mono_set (Set.Ioc_subset_Ioc_left ha)\n have hs_mono : StrictMono (fun n => ∑ i ∈ Finset.range n, X i ω) := by\n apply strictMono_nat_of_lt_succ\n intro n\n rw [Finset.sum_range_succ]\n linarith [hpos n ω]\n have hspos : ∀ n : ℕ, 1 ≤ n → 0 < ∑ i ∈ Finset.range n, X i ω := by\n intro n hn\n have := hs_mono (show 0 < n by omega)\n simpa using this\n have hgsn : ∀ n : ℕ, (∫ τ in (0:ℝ)..(∑ i ∈ Finset.range n, X i ω), R τ ω)\n = ∑ i ∈ Finset.range n, Rcyc i ω := by\n intro n\n have hint : ∀ k, k < n → IntervalIntegrable (fun τ => R τ ω) volume\n (∑ i ∈ Finset.range k, X i ω) (∑ i ∈ Finset.range (k + 1), X i ω) := by\n intro k hk\n apply hII\n · rcases Nat.eq_zero_or_pos k with hk0 | hk0\n · simp [hk0]\n · exact le_of_lt (hspos k hk0)\n · rw [Finset.sum_range_succ]\n linarith [hpos k ω]\n have key := intervalIntegral.sum_integral_adjacent_intervals\n (a := fun m => ∑ i ∈ Finset.range m, X i ω) hint\n simp only [Finset.sum_range_zero] at key\n rw [← key]\n apply Finset.sum_congr rfl\n intro k hk\n rw [hlink k ω, hS k ω, hS (k + 1) ω]\n have hg_mono : ∀ a b : ℝ, 0 ≤ a → a ≤ b →\n (∫ τ in (0:ℝ)..a, R τ ω) ≤ (∫ τ in (0:ℝ)..b, R τ ω) := by\n intro a b ha hab\n have h1 : IntervalIntegrable (fun τ => R τ ω) volume 0 a := hII 0 a le_rfl ha\n have h2 : IntervalIntegrable (fun τ => R τ ω) volume a b := hII a b ha hab\n have hadd := intervalIntegral.integral_add_adjacent_intervals h1 h2\n have hnn : 0 ≤ ∫ τ in a..b, R τ ω :=\n intervalIntegral.integral_nonneg hab (fun x _ => hRnn x ω)\n linarith [hadd, hnn]\n have hg_nonneg : ∀ t : ℝ, 0 ≤ t → 0 ≤ (∫ τ in (0:ℝ)..t, R τ ω) := by\n intro t ht\n exact intervalIntegral.integral_nonneg ht (fun x _ => hRnn x ω)\n have hquot : Tendsto\n (fun n => (∑ i ∈ Finset.range n, Rcyc i ω) / (∑ i ∈ Finset.range n, X i ω))\n atTop (𝓝 (r / Xbar)) := by\n have hdiv := hSR.div hSX (ne_of_gt hXpos)\n refine hdiv.congr' ?_\n filter_upwards [eventually_ge_atTop 1] with n hn\n have hn0 : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (by omega)\n exact div_div_div_cancel_right₀ hn0 _ _\n have hratio : Tendsto\n (fun n => (∫ τ in (0:ℝ)..(∑ i ∈ Finset.range n, X i ω), R τ ω)\n / (∑ i ∈ Finset.range n, X i ω)) atTop (𝓝 (r / Xbar)) :=\n hquot.congr (fun n => by rw [← hgsn n])\n have final : Tendsto (fun t => (∫ τ in (0:ℝ)..t, R τ ω) / t) atTop (𝓝 (r / Xbar)) := by\n apply renewal_squeeze (fun t => ∫ τ in (0:ℝ)..t, R τ ω)\n (fun n => ∑ i ∈ Finset.range n, X i ω) (r / Xbar) Xbar hXpos\n · simp\n · exact hs_mono\n · exact hSX\n · exact hg_mono\n · exact hg_nonneg\n · exact hratio\n exact final.congr (fun t => by ring)", "time_seconds": 1497, "topic": "Renewal processes" }, { "question_id": 266, "problem_name": "stochastic_processes_markov_modulated_walk_martingale", "informal_statement": "Let (S_n) be a Markov-modulated random walk whose increments depend on an underlying finite-state Markov chain, with the per-state generating functions combined into the matrix [A(r)] having Perron eigenvalue ρ(r), and form the product-form process based on the corresponding right eigenvector.\n\nShow that this product-form process is a martingale.", "lean_statement": "theorem q266_markov_modulated_product_martingale\n (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ]\n (k : ℕ) [NeZero k]\n (A : Matrix (Fin k) (Fin k) ℝ) (ρ : ℝ) (hρ : 0 < ρ)\n (ν : Fin k → ℝ) (hν : ∀ i, 0 < ν i)\n (heig : ∀ i, ∑ j, A i j * ν j = ρ * ν i)\n (r : ℝ) (X : ℕ → Ω → Fin k) (S : ℕ → Ω → ℝ)\n (hXadapt : StronglyAdapted ℱ X) (hSadapt : StronglyAdapted ℱ S)\n (hmodelInt : ∀ n j, Integrable (fun ω => Real.exp (r * (S (n + 1) ω - S n ω)) *\n (if X (n + 1) ω = j then (1 : ℝ) else 0)) μ)\n (M : ℕ → Ω → ℝ) (hM : ∀ n ω, M n ω = Real.exp (r * S n ω) * ν (X n ω) / ρ ^ n)\n (hadapt : StronglyAdapted ℱ M) (hint : ∀ n, Integrable (M n) μ)\n (hmodel : ∀ n j,\n μ[fun ω => Real.exp (r * (S (n + 1) ω - S n ω)) * (if X (n + 1) ω = j then (1 : ℝ) else 0)\n | ℱ n]\n =ᵐ[μ] fun ω => A (X n ω) j) :\n Martingale M ℱ μ := by\n sorry", "true_to_source": "abstract", "proof": "theorem q266_markov_modulated_product_martingale\n (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0) [SigmaFiniteFiltration μ ℱ]\n (k : ℕ) [NeZero k]\n (A : Matrix (Fin k) (Fin k) ℝ) (ρ : ℝ) (hρ : 0 < ρ)\n (ν : Fin k → ℝ) (hν : ∀ i, 0 < ν i)\n (heig : ∀ i, ∑ j, A i j * ν j = ρ * ν i)\n (r : ℝ) (X : ℕ → Ω → Fin k) (S : ℕ → Ω → ℝ)\n (hXadapt : StronglyAdapted ℱ X) (hSadapt : StronglyAdapted ℱ S)\n (hmodelInt : ∀ n j, Integrable (fun ω => Real.exp (r * (S (n + 1) ω - S n ω)) *\n (if X (n + 1) ω = j then (1 : ℝ) else 0)) μ)\n (M : ℕ → Ω → ℝ) (hM : ∀ n ω, M n ω = Real.exp (r * S n ω) * ν (X n ω) / ρ ^ n)\n (hadapt : StronglyAdapted ℱ M) (hint : ∀ n, Integrable (M n) μ)\n (hmodel : ∀ n j,\n μ[fun ω => Real.exp (r * (S (n + 1) ω - S n ω)) * (if X (n + 1) ω = j then (1 : ℝ) else 0)\n | ℱ n]\n =ᵐ[μ] fun ω => A (X n ω) j) :\n Martingale M ℱ μ := by\n apply martingale_nat hadapt hint\n intro n\n -- Basic facts about ρ.\n have hρ0 : ρ ≠ 0 := hρ.ne'\n have hρn0 : ρ ^ n ≠ 0 := pow_ne_zero n hρ0\n -- The ℱ n-measurable \"current\" exponential factor.\n have hSsm : StronglyMeasurable[ℱ n] (S n) := hSadapt n\n have hEexp : StronglyMeasurable[ℱ n] (fun ω => Real.exp (r * S n ω)) :=\n Real.continuous_exp.comp_stronglyMeasurable (hSsm.const_mul r)\n -- The per-state model function `F j` and the ℱ n-measurable coefficient `c j`.\n set F : Fin k → Ω → ℝ :=\n fun j ω => Real.exp (r * (S (n + 1) ω - S n ω)) * (if X (n + 1) ω = j then (1 : ℝ) else 0)\n with hFdef\n set c : Fin k → Ω → ℝ :=\n fun j ω => Real.exp (r * S n ω) / ρ ^ (n + 1) * ν j with hcdef\n -- Restate the hypotheses in terms of `F`.\n have hmodelF : ∀ j, μ[F j | ℱ n] =ᵐ[μ] fun ω => A (X n ω) j := by\n intro j; simp only [hFdef]; exact hmodel n j\n have hmodelIntF : ∀ j, Integrable (F j) μ := by\n intro j; simp only [hFdef]; exact hmodelInt n j\n -- `c j` is strongly ℱ n-measurable.\n have hcsm : ∀ j, StronglyMeasurable[ℱ n] (c j) := by\n intro j; simp only [hcdef]; exact (hEexp.div stronglyMeasurable_const).mul_const _\n -- Each product term collapses to a masked copy of `M (n+1)`.\n have hterm : ∀ j ω, c j ω * F j ω = if X (n + 1) ω = j then M (n + 1) ω else 0 := by\n intro j ω\n simp only [hcdef, hFdef]\n have hexp : Real.exp (r * S n ω) * Real.exp (r * (S (n + 1) ω - S n ω))\n = Real.exp (r * S (n + 1) ω) := by\n rw [← Real.exp_add]; congr 1; ring\n split_ifs with h\n · rw [hM (n + 1) ω, h, mul_one, ← hexp]; ring\n · ring\n -- Integrability of each product term (dominated by `M (n+1)`).\n have hint_prod : ∀ j, Integrable (fun ω => c j ω * F j ω) μ := by\n intro j\n apply Integrable.mono' (hint (n + 1)).norm\n · exact (((hcsm j).mono (ℱ.le n)).aestronglyMeasurable).mul (hmodelIntF j).aestronglyMeasurable\n · refine ae_of_all μ (fun ω => ?_)\n rw [hterm j ω]\n split_ifs with h\n · exact le_refl _\n · rw [norm_zero]; exact norm_nonneg _\n -- `M (n+1)` is the (pointwise) sum of the product terms.\n have hMsum_pointwise : ∀ ω, M (n + 1) ω = ∑ j, c j ω * F j ω := by\n intro ω\n rw [Finset.sum_congr rfl (fun j _ => hterm j ω), Finset.sum_ite_eq]\n simp\n have hMeq : M (n + 1) =ᵐ[μ] ∑ j, c j * F j := by\n refine ae_of_all μ (fun ω => ?_)\n rw [Finset.sum_apply]\n simp only [Pi.mul_apply]\n exact hMsum_pointwise ω\n -- Pull the conditional expectation through the sum and the ℱ n-measurable coefficients.\n have step1 : μ[M (n + 1) | ℱ n] =ᵐ[μ] μ[∑ j, c j * F j | ℱ n] := condExp_congr_ae hMeq\n have step2 : μ[∑ j, c j * F j | ℱ n] =ᵐ[μ] ∑ j, μ[c j * F j | ℱ n] :=\n condExp_finsetSum (fun j _ => hint_prod j) (ℱ n)\n have step3 : ∀ j, μ[c j * F j | ℱ n] =ᵐ[μ] c j * μ[F j | ℱ n] := fun j =>\n condExp_mul_of_stronglyMeasurable_left (hcsm j) (hint_prod j) (hmodelIntF j)\n have step23 : μ[∑ j, c j * F j | ℱ n] =ᵐ[μ] ∑ j, c j * μ[F j | ℱ n] := by\n refine step2.trans ?_\n have hall : ∀ᵐ ω ∂μ, ∀ j, μ[c j * F j | ℱ n] ω = (c j * μ[F j | ℱ n]) ω := by\n rw [ae_all_iff]; exact step3\n filter_upwards [hall] with ω hω\n rw [Finset.sum_apply, Finset.sum_apply]\n exact Finset.sum_congr rfl (fun j _ => hω j)\n have hcondsum : μ[M (n + 1) | ℱ n] =ᵐ[μ] ∑ j, c j * μ[F j | ℱ n] := step1.trans step23\n -- Apply the model identity and the eigen-relation to identify the limit with `M n`.\n have hmodel_all : ∀ᵐ ω ∂μ, ∀ j, μ[F j | ℱ n] ω = A (X n ω) j := by\n rw [ae_all_iff]; exact hmodelF\n have hfinal : μ[M (n + 1) | ℱ n] =ᵐ[μ] M n := by\n filter_upwards [hcondsum, hmodel_all] with ω hsum hmod\n rw [hsum, Finset.sum_apply]\n simp only [Pi.mul_apply, hmod, hcdef]\n have hcompute : ∑ j, Real.exp (r * S n ω) / ρ ^ (n + 1) * ν j * A (X n ω) j\n = Real.exp (r * S n ω) / ρ ^ (n + 1) * (ρ * ν (X n ω)) := by\n rw [← heig (X n ω), Finset.mul_sum]\n exact Finset.sum_congr rfl (fun j _ => by ring)\n rw [hcompute, hM n ω, pow_succ]\n field_simp\n exact hfinal.symm", "time_seconds": 870, "topic": "Martingales & stopping" }, { "question_id": 288, "problem_name": "stochastic_processes_continous_markov_holding_time", "informal_statement": "Let S_x be the measurable, almost surely finite holding time at a nonabsorbing state of a time-homogeneous continuous-time Markov chain. Its survival function is memoryless and P(S_x>0)>0.\n\nShow that S_x has an exponential distribution with a strictly positive rate.", "lean_statement": "theorem q288_ctmc_holding_time (μ : Measure Ω) [IsProbabilityMeasure μ] (Sx : Ω → ℝ) (hSmeas : Measurable Sx)\n (hpos : ∀ ω, 0 ≤ Sx ω) (hnt : 0 < (μ {ω | 0 < Sx ω}).toReal)\n (hmem : ∀ s t, 0 ≤ s → 0 ≤ t →\n (μ {ω | s + t < Sx ω}).toReal = (μ {ω | s < Sx ω}).toReal * (μ {ω | t < Sx ω}).toReal) :\n ∃ rate : ℝ, 0 < rate ∧ HasLaw Sx (expMeasure rate) μ := by sorry", "true_to_source": "abstract", "proof": "private lemma memoryless_survival_rpow {G : ℝ → ℝ} {b : ℝ} (hb : 0 < b)\n (hanti : Antitone G) (hGnn : ∀ t, 0 ≤ G t)\n (hmul : ∀ s t, 0 ≤ s → 0 ≤ t → G (s + t) = G s * G t)\n (hG0 : G 0 = 1) (hG1 : G 1 = b) :\n ∀ t, 0 < t → G t = b ^ t := by\n have hpow : ∀ (n : ℕ) (t : ℝ), 0 ≤ t → G ((n : ℝ) * t) = (G t) ^ n := by\n intro n\n induction n with\n | zero => intro t ht; simp only [Nat.cast_zero, zero_mul, pow_zero, hG0]\n | succ k ih =>\n intro t ht\n have hkt : (0 : ℝ) ≤ (k : ℝ) * t := mul_nonneg (Nat.cast_nonneg k) ht\n have hcc : ((k + 1 : ℕ) : ℝ) * t = (k : ℝ) * t + t := by push_cast; ring\n rw [hcc, hmul _ _ hkt ht, ih t ht, pow_succ]\n have hGnat : ∀ n : ℕ, G ((n : ℝ)) = b ^ n := by\n intro n\n have h := hpow n 1 zero_le_one\n rwa [mul_one, hG1] at h\n have hfrac : ∀ (m n : ℕ), 0 < n → G ((m : ℝ) / n) = b ^ ((m : ℝ) / n) := by\n intro m n hn\n have hqnn : (0 : ℝ) ≤ (m : ℝ) / n := by positivity\n have key : (G ((m : ℝ) / n)) ^ n = b ^ m := by\n have h1 := hpow n ((m : ℝ) / n) hqnn\n rw [show (n : ℝ) * ((m : ℝ) / n) = (m : ℝ) by field_simp] at h1\n rw [hGnat m] at h1\n exact h1.symm\n have hx : 0 ≤ G ((m : ℝ) / n) := hGnn _\n calc G ((m : ℝ) / n)\n = (G ((m : ℝ) / n) ^ n) ^ ((n : ℝ)⁻¹) := (Real.pow_rpow_inv_natCast hx hn.ne').symm\n _ = (b ^ m) ^ ((n : ℝ)⁻¹) := by rw [key]\n _ = (b ^ (m : ℝ)) ^ ((n : ℝ)⁻¹) := by rw [Real.rpow_natCast]\n _ = b ^ ((m : ℝ) * (n : ℝ)⁻¹) := (Real.rpow_mul hb.le _ _).symm\n _ = b ^ ((m : ℝ) / n) := by rw [div_eq_mul_inv]\n have hrat : ∀ q : ℚ, 0 ≤ q → G ((q : ℝ)) = b ^ (q : ℝ) := by\n intro q hq\n have hnum : 0 ≤ q.num := Rat.num_nonneg.mpr hq\n have hcast : (q.num.toNat : ℝ) / (q.den : ℝ) = (q : ℝ) := by\n rw [Rat.cast_def]\n congr 1\n exact_mod_cast Int.toNat_of_nonneg hnum\n rw [← hcast]\n exact hfrac q.num.toNat q.den q.den_pos\n have hcont : Continuous (fun x : ℝ => b ^ x) := by\n simp only [Real.rpow_def_of_pos hb]\n fun_prop\n intro t ht\n apply le_antisymm\n · by_contra hcon\n push_neg at hcon\n have hop : IsOpen ((fun x : ℝ => b ^ x) ⁻¹' Set.Iio (G t)) := isOpen_Iio.preimage hcont\n have hmem2 : ((fun x : ℝ => b ^ x) ⁻¹' Set.Iio (G t)) ∩ Set.Ioi 0 ∈ nhds t :=\n (hop.inter isOpen_Ioi).mem_nhds ⟨hcon, ht⟩\n rw [mem_nhds_iff_exists_Ioo_subset] at hmem2\n obtain ⟨l, u, htlu, hsub⟩ := hmem2\n obtain ⟨q, hlq, hqt⟩ := exists_rat_btwn htlu.1\n have hqmem : (q : ℝ) ∈ Set.Ioo l u := ⟨hlq, lt_trans hqt htlu.2⟩\n have hqin := hsub hqmem\n have hbq : b ^ (q : ℝ) < G t := hqin.1\n have hq0 : (0 : ℝ) ≤ (q : ℝ) := le_of_lt hqin.2\n have hGq : G ((q : ℝ)) = b ^ (q : ℝ) := hrat q (by exact_mod_cast hq0)\n have hle : G t ≤ G ((q : ℝ)) := hanti hqt.le\n rw [hGq] at hle\n linarith\n · by_contra hcon\n push_neg at hcon\n have hop : IsOpen ((fun x : ℝ => b ^ x) ⁻¹' Set.Ioi (G t)) := isOpen_Ioi.preimage hcont\n have hmem2 : ((fun x : ℝ => b ^ x) ⁻¹' Set.Ioi (G t)) ∈ nhds t := hop.mem_nhds hcon\n rw [mem_nhds_iff_exists_Ioo_subset] at hmem2\n obtain ⟨l, u, htlu, hsub⟩ := hmem2\n obtain ⟨q, htq, hqu⟩ := exists_rat_btwn htlu.2\n have hqmem : (q : ℝ) ∈ Set.Ioo l u := ⟨lt_trans htlu.1 htq, hqu⟩\n have hqin := hsub hqmem\n have hbq : G t < b ^ (q : ℝ) := hqin\n have hq0 : (0 : ℝ) ≤ (q : ℝ) := le_of_lt (lt_trans ht htq)\n have hGq : G ((q : ℝ)) = b ^ (q : ℝ) := hrat q (by exact_mod_cast hq0)\n have hle : G ((q : ℝ)) ≤ G t := hanti htq.le\n rw [hGq] at hle\n linarith\n\ntheorem q288_ctmc_holding_time (μ : Measure Ω) [IsProbabilityMeasure μ] (Sx : Ω → ℝ) (hSmeas : Measurable Sx)\n (hpos : ∀ ω, 0 ≤ Sx ω) (hnt : 0 < (μ {ω | 0 < Sx ω}).toReal)\n (hmem : ∀ s t, 0 ≤ s → 0 ≤ t →\n (μ {ω | s + t < Sx ω}).toReal = (μ {ω | s < Sx ω}).toReal * (μ {ω | t < Sx ω}).toReal) :\n ∃ rate : ℝ, 0 < rate ∧ HasLaw Sx (expMeasure rate) μ := by\n have hGnn : ∀ t : ℝ, 0 ≤ (μ {ω | t < Sx ω}).toReal := fun t => ENNReal.toReal_nonneg\n have hanti : Antitone (fun t : ℝ => (μ {ω | t < Sx ω}).toReal) := by\n intro a b hab\n exact ENNReal.toReal_mono (measure_ne_top μ _)\n (measure_mono (fun ω hω => lt_of_le_of_lt hab hω))\n have hG0 : (μ {ω | (0 : ℝ) < Sx ω}).toReal = 1 := by\n have h2 := hmem 0 0 le_rfl le_rfl\n rw [add_zero] at h2\n have hcne : (μ {ω | (0 : ℝ) < Sx ω}).toReal ≠ 0 := ne_of_gt hnt\n have h3 : (μ {ω | (0 : ℝ) < Sx ω}).toReal * (μ {ω | (0 : ℝ) < Sx ω}).toReal\n = (μ {ω | (0 : ℝ) < Sx ω}).toReal * 1 := by rw [mul_one]; exact h2.symm\n exact mul_left_cancel₀ hcne h3\n have hGnat : ∀ n : ℕ,\n (μ {ω | (n : ℝ) < Sx ω}).toReal = ((μ {ω | (1 : ℝ) < Sx ω}).toReal) ^ n := by\n intro n\n induction n with\n | zero => simpa using hG0\n | succ k ih =>\n have hc : ((k + 1 : ℕ) : ℝ) = (k : ℝ) + 1 := by push_cast; ring\n have hm := hmem (k : ℝ) 1 (Nat.cast_nonneg k) zero_le_one\n rw [hc, hm, ih, pow_succ]\n have hb_le_one : (μ {ω | (1 : ℝ) < Sx ω}).toReal ≤ 1 := by\n simpa using ENNReal.toReal_mono ENNReal.one_ne_top\n (prob_le_one : μ {ω | (1 : ℝ) < Sx ω} ≤ 1)\n have hb_pos : 0 < (μ {ω | (1 : ℝ) < Sx ω}).toReal := by\n rcases (hGnn 1).lt_or_eq with h | h\n · exact h\n · exfalso\n have hzero : ∀ n : ℕ, (μ {ω | (1 / 2 : ℝ) ^ n < Sx ω}).toReal = 0 := by\n intro n\n induction n with\n | zero => simpa using h.symm\n | succ k ih =>\n have hsplit : (1 / 2 : ℝ) ^ k = (1 / 2 : ℝ) ^ (k + 1) + (1 / 2 : ℝ) ^ (k + 1) := by\n rw [pow_succ]; ring\n have hpos' : (0 : ℝ) ≤ (1 / 2 : ℝ) ^ (k + 1) := by positivity\n have hm := hmem ((1 / 2 : ℝ) ^ (k + 1)) ((1 / 2 : ℝ) ^ (k + 1)) hpos' hpos'\n rw [← hsplit, ih] at hm\n exact (mul_self_eq_zero).1 hm.symm\n have hmono : Monotone (fun n : ℕ => {ω | (1 / 2 : ℝ) ^ n < Sx ω}) := by\n intro i j hij ω hω\n simp only [Set.mem_setOf_eq] at hω ⊢\n exact lt_of_le_of_lt (pow_le_pow_of_le_one (by norm_num) (by norm_num) hij) hω\n have hunion : (⋃ n : ℕ, {ω | (1 / 2 : ℝ) ^ n < Sx ω}) = {ω | (0 : ℝ) < Sx ω} := by\n ext ω\n simp only [Set.mem_iUnion, Set.mem_setOf_eq]\n constructor\n · rintro ⟨n, hn⟩\n exact lt_trans (by positivity) hn\n · intro hω\n obtain ⟨n, hn⟩ := exists_pow_lt_of_lt_one hω (by norm_num : (1 / 2 : ℝ) < 1)\n exact ⟨n, hn⟩\n have htend := tendsto_measure_iUnion_atTop (μ := μ) hmono\n rw [hunion] at htend\n have hμ1 : μ {ω | (0 : ℝ) < Sx ω} = 1 := by\n rw [← ENNReal.ofReal_toReal (measure_ne_top μ {ω | (0 : ℝ) < Sx ω}), hG0,\n ENNReal.ofReal_one]\n rw [hμ1] at htend\n have htendR : Tendsto (fun n : ℕ => (μ {ω | (1 / 2 : ℝ) ^ n < Sx ω}).toReal)\n atTop (𝓝 (1 : ℝ)) := by\n have h1 : (1 : ℝ) = (1 : ℝ≥0∞).toReal := by simp\n rw [h1]\n exact (ENNReal.tendsto_toReal ENNReal.one_ne_top).comp htend\n rw [tendsto_congr hzero] at htendR\n exact absurd (tendsto_nhds_unique htendR tendsto_const_nhds) (by norm_num)\n have hb_lt_one : (μ {ω | (1 : ℝ) < Sx ω}).toReal < 1 := by\n rcases lt_or_eq_of_le hb_le_one with h | h\n · exact h\n · exfalso\n have hanti' : Antitone (fun n : ℕ => {ω | (n : ℝ) < Sx ω}) := by\n intro i j hij ω hω\n simp only [Set.mem_setOf_eq] at hω ⊢\n exact lt_of_le_of_lt (by exact_mod_cast hij) hω\n have hinter : (⋂ n : ℕ, {ω | (n : ℝ) < Sx ω}) = ∅ := by\n rw [Set.eq_empty_iff_forall_notMem]\n intro ω hω\n rw [Set.mem_iInter] at hω\n obtain ⟨n, hn⟩ := exists_nat_gt (Sx ω)\n exact absurd (hω n) (not_lt.2 (le_of_lt hn))\n have htend := tendsto_measure_iInter_atTop (μ := μ)\n (fun n => (measurableSet_lt measurable_const hSmeas).nullMeasurableSet)\n hanti' ⟨0, measure_ne_top μ _⟩\n rw [hinter, measure_empty] at htend\n have htendR : Tendsto (fun n : ℕ => (μ {ω | (n : ℝ) < Sx ω}).toReal)\n atTop (𝓝 (0 : ℝ)) := by\n have h0' : (0 : ℝ) = (0 : ℝ≥0∞).toReal := by simp\n rw [h0']\n exact (ENNReal.tendsto_toReal ENNReal.zero_ne_top).comp htend\n have hconst : ∀ n : ℕ, (μ {ω | (n : ℝ) < Sx ω}).toReal = 1 := by\n intro n; rw [hGnat n, h, one_pow]\n rw [tendsto_congr hconst] at htendR\n exact absurd (tendsto_nhds_unique htendR tendsto_const_nhds) (by norm_num)\n have hGrpow : ∀ t : ℝ, 0 < t →\n (μ {ω | t < Sx ω}).toReal = ((μ {ω | (1 : ℝ) < Sx ω}).toReal) ^ t :=\n memoryless_survival_rpow (G := fun t : ℝ => (μ {ω | t < Sx ω}).toReal)\n hb_pos hanti hGnn hmem hG0 rfl\n have hrpos : 0 < - Real.log ((μ {ω | (1 : ℝ) < Sx ω}).toReal) := by\n have := Real.log_neg hb_pos hb_lt_one\n linarith\n refine ⟨- Real.log ((μ {ω | (1 : ℝ) < Sx ω}).toReal), hrpos, ?_⟩\n set r := - Real.log ((μ {ω | (1 : ℝ) < Sx ω}).toReal) with hr\n have hGexp : ∀ a : ℝ, 0 ≤ a → (μ {ω | a < Sx ω}).toReal = Real.exp (-(r * a)) := by\n intro a ha\n rcases eq_or_lt_of_le ha with h0 | h0\n · rw [← h0]; simp only [mul_zero, neg_zero, Real.exp_zero]; exact hG0\n · rw [hGrpow a h0, Real.rpow_def_of_pos hb_pos]\n congr 1\n rw [hr]; ring\n haveI : IsProbabilityMeasure (μ.map Sx) := Measure.isProbabilityMeasure_map hSmeas.aemeasurable\n haveI : IsProbabilityMeasure (expMeasure r) := isProbabilityMeasure_expMeasure hrpos\n refine ⟨hSmeas.aemeasurable, ?_⟩\n refine MeasureTheory.Measure.ext_of_Iic (μ.map Sx) (expMeasure r) (fun a => ?_)\n rw [Measure.map_apply hSmeas measurableSet_Iic,\n ← ProbabilityTheory.ofReal_cdf (expMeasure r) a, cdf_expMeasure_eq hrpos a]\n have hpre : Sx ⁻¹' Set.Iic a = {ω | Sx ω ≤ a} := rfl\n rw [hpre]\n split_ifs with ha\n · have hcompl : {ω | Sx ω ≤ a} = {ω | a < Sx ω}ᶜ := by\n ext ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt]\n rw [hcompl, measure_compl (measurableSet_lt measurable_const hSmeas) (measure_ne_top μ _),\n measure_univ, ← hGexp a ha, ENNReal.ofReal_sub _ ENNReal.toReal_nonneg, ENNReal.ofReal_one,\n ENNReal.ofReal_toReal (measure_ne_top μ _)]\n · have hempty : {ω | Sx ω ≤ a} = ∅ := by\n rw [Set.eq_empty_iff_forall_notMem]\n intro ω hω\n simp only [Set.mem_setOf_eq] at hω\n exact absurd (lt_of_lt_of_le (not_le.1 ha) (hpos ω)) (not_lt.2 hω)\n rw [hempty, measure_empty, ENNReal.ofReal_zero]", "time_seconds": 1666, "topic": "Continuous-time Markov & queues" }, { "question_id": 346, "problem_name": "stochastic_processes_nonhomogeneous_time_change", "informal_statement": "Let N be a nonhomogeneous Poisson counting process on [0,∞), starting at zero, with continuous strictly increasing mean function m, m(0)=0, and m(t)→∞. In the rate formulation, m(t)=∫_0^t λ(u)du.\n\nShow that there exists a rate-one Poisson process N* on the same probability space such that N(t)=N*(m(t)) for all t≥0. The unbounded clock ensures its inverse is defined on the whole nonnegative half-line.", "lean_statement": "theorem q346_nonhomogeneous_time_change (μ : Measure Ω) [IsProbabilityMeasure μ]\n (N : ℝ≥0 → Ω → ℕ) (hNcount : IsCountingProcess μ N)\n (hindep : HasIndepIncrements N μ)\n (m : ℝ≥0 → ℝ≥0) (hm0 : m 0 = 0) (hmcont : Continuous m)\n (hmmono : StrictMono m) (hmunbounded : Tendsto m atTop atTop)\n (hincr : ∀ a b : ℝ≥0, a ≤ b →\n HasLaw (fun ω => N b ω - N a ω) (poissonMeasure (m b - m a)) μ) :\n ∃ Nstar : ℝ≥0 → Ω → ℕ,\n IsPoissonProcess μ 1 Nstar ∧ ∀ t ω, N t ω = Nstar (m t) ω := by\n sorry\n", "true_to_source": "abstract", "proof": "theorem q346_nonhomogeneous_time_change (μ : Measure Ω) [IsProbabilityMeasure μ]\n (N : ℝ≥0 → Ω → ℕ) (hNcount : IsCountingProcess μ N)\n (hindep : HasIndepIncrements N μ)\n (m : ℝ≥0 → ℝ≥0) (hm0 : m 0 = 0) (hmcont : Continuous m)\n (hmmono : StrictMono m) (hmunbounded : Tendsto m atTop atTop)\n (hincr : ∀ a b : ℝ≥0, a ≤ b →\n HasLaw (fun ω => N b ω - N a ω) (poissonMeasure (m b - m a)) μ) :\n ∃ Nstar : ℝ≥0 → Ω → ℕ,\n IsPoissonProcess μ 1 Nstar ∧ ∀ t ω, N t ω = Nstar (m t) ω := by\n -- `m` is surjective onto `[0,∞)` by the intermediate value theorem and unboundedness.\n have hsurj : Function.Surjective m := by\n intro y\n obtain ⟨b, hb⟩ := (hmunbounded.eventually_ge_atTop y).exists\n have h0 : m 0 ≤ y := by rw [hm0]; exact zero_le'\n have hmem : y ∈ Set.Icc (m 0) (m b) := ⟨h0, hb⟩\n obtain ⟨t, _, ht⟩ :=\n intermediate_value_Icc (zero_le' : (0 : ℝ≥0) ≤ b) hmcont.continuousOn hmem\n exact ⟨t, ht⟩\n -- The clock `m` as an order isomorphism, together with its inverse `e.symm`.\n let e : ℝ≥0 ≃o ℝ≥0 := StrictMono.orderIsoOfSurjective m hmmono hsurj\n have he : ∀ t, e t = m t := fun t => rfl\n have hsymm_m : ∀ t, e.symm (m t) = t := fun t => by\n rw [← he t]; exact e.symm_apply_apply t\n have hm_symm : ∀ s, m (e.symm s) = s := fun s => by\n rw [← he (e.symm s)]; exact e.apply_symm_apply s\n have hmono_symm : Monotone e.symm := e.symm.monotone\n have hcont_symm : Continuous e.symm := e.symm.continuous\n have he0 : e.symm 0 = 0 := by\n have h := hsymm_m 0; rwa [hm0] at h\n -- The rate-one process is `N` reparametrised by the inverse clock.\n refine ⟨fun s ω => N (e.symm s) ω, ⟨⟨?_, ?_, ?_⟩, ?_, ?_⟩, ?_⟩\n · -- measurability of each time slice\n intro s\n exact hNcount.1 (e.symm s)\n · -- starts at zero almost surely\n show (fun ω => N (e.symm 0) ω) =ᵐ[μ] fun _ => 0\n rw [he0]; exact hNcount.2.1\n · -- monotone, right-continuous paths almost surely\n filter_upwards [hNcount.2.2] with ω hω\n refine ⟨hω.1.comp hmono_symm, ?_⟩\n intro t\n exact (hω.2 (e.symm t)).comp hcont_symm.continuousWithinAt (fun _ hs => hmono_symm hs)\n · -- independent increments transfer through the monotone reparametrisation\n intro n t ht\n exact hindep n (fun i => e.symm (t i)) (hmono_symm.comp ht)\n · -- increment law: a `[a,b]` increment matches an `[m⁻¹a, m⁻¹b]` increment of `N`\n intro a b hab\n have key : poissonMeasure (1 * (b - a))\n = poissonMeasure (m (e.symm b) - m (e.symm a)) := by\n rw [one_mul, hm_symm, hm_symm]\n rw [key]\n exact hincr (e.symm a) (e.symm b) (hmono_symm hab)\n · -- the time-change identity `N t = Nstar (m t)`\n intro t ω\n show N t ω = N (e.symm (m t)) ω\n rw [hsymm_m t]", "time_seconds": 674, "topic": "Poisson processes" }, { "question_id": 368, "problem_name": "stochastic_processes_convex_distance_hull_characterization", "informal_statement": "Let D^c_A be the convex distance associated with a set A in a product space, and let V_A(x) be the convex hull of the sign vectors {(1{x_i ≠ y_i})_i : y ∈ A}.\n\nShow that D^c_A(x) = d(0, V_A(x)) = inf_{y ∈ V_A(x)} |y|.", "lean_statement": "theorem q368_convex_distance_eq_infdist\n (n : ℕ) (α : Fin n → Type*)\n (A : Set (∀ i, α i)) (hAne : A.Nonempty)\n (x : ∀ i, α i)\n (S : Set (EuclideanSpace ℝ (Fin n)))\n (hS : S = {v | ∃ y ∈ A, ∀ i, v i = if x i = y i then (0 : ℝ) else 1}) :\n (⨆ a : {a : EuclideanSpace ℝ (Fin n) // ‖a‖ = 1},\n ⨅ y : A, ∑ i, a.val i * (if x i = y.val i then (0 : ℝ) else 1))\n = Metric.infDist (0 : EuclideanSpace ℝ (Fin n)) (convexHull ℝ S) ∧\n Metric.infDist (0 : EuclideanSpace ℝ (Fin n)) (convexHull ℝ S)\n = ⨅ v : convexHull ℝ S, ‖v.val‖ := by\n sorry", "true_to_source": "abstract", "proof": "private lemma inner_eq_sum' {n : ℕ} (a v : EuclideanSpace ℝ (Fin n)) :\n inner ℝ a v = ∑ i, a i * v i := by\n simp only [PiLp.inner_apply, RCLike.inner_apply, conj_trivial]\n exact Finset.sum_congr rfl fun i _ => mul_comm _ _\n\ntheorem q368_convex_distance_eq_infdist\n (n : ℕ) (α : Fin n → Type*)\n (A : Set (∀ i, α i)) (hAne : A.Nonempty)\n (x : ∀ i, α i)\n (S : Set (EuclideanSpace ℝ (Fin n)))\n (hS : S = {v | ∃ y ∈ A, ∀ i, v i = if x i = y i then (0 : ℝ) else 1}) :\n (⨆ a : {a : EuclideanSpace ℝ (Fin n) // ‖a‖ = 1},\n ⨅ y : A, ∑ i, a.val i * (if x i = y.val i then (0 : ℝ) else 1))\n = Metric.infDist (0 : EuclideanSpace ℝ (Fin n)) (convexHull ℝ S) ∧\n Metric.infDist (0 : EuclideanSpace ℝ (Fin n)) (convexHull ℝ S)\n = ⨅ v : convexHull ℝ S, ‖v.val‖ := by\n classical\n haveI : Nonempty ↥A := hAne.to_subtype\n set φ : (∀ i, α i) → EuclideanSpace ℝ (Fin n) :=\n fun y => WithLp.toLp 2 (fun i => if x i = y i then (0 : ℝ) else 1) with hφ\n have hφi : ∀ y i, (φ y) i = if x i = y i then (0 : ℝ) else 1 := fun y i => by simp [hφ]\n have hterm : ∀ (a : EuclideanSpace ℝ (Fin n)) (y : ∀ i, α i),\n (∑ i, a i * (if x i = y i then (0 : ℝ) else 1)) = inner ℝ a (φ y) := by\n intro a y; rw [inner_eq_sum']\n have hφmem : ∀ y, y ∈ A → φ y ∈ S := by\n intro y hy; rw [hS]; exact ⟨y, hy, fun i => hφi y i⟩\n have hSne : S.Nonempty := by\n obtain ⟨y₀, hy₀⟩ := hAne; exact ⟨φ y₀, hφmem y₀ hy₀⟩\n have hSfin : S.Finite := by\n apply Set.Finite.subset (Set.finite_range\n (fun b : Fin n → Bool => WithLp.toLp 2 ((fun i => if b i then (1 : ℝ) else 0))))\n rw [hS]\n rintro v ⟨y, hy, hvy⟩\n refine ⟨fun i => if x i = y i then false else true, ?_⟩\n ext i\n by_cases h : x i = y i <;> simp [h, hvy i]\n have hCne : (convexHull ℝ S).Nonempty := hSne.mono (subset_convexHull ℝ S)\n have hCconv : Convex ℝ (convexHull ℝ S) := convex_convexHull ℝ S\n have hCcomplete : IsComplete (convexHull ℝ S) :=\n (Set.Finite.isCompact_convexHull (𝕜 := ℝ) hSfin).isComplete\n obtain ⟨p, hpC, hpeq⟩ := exists_norm_eq_iInf_of_complete_convex hCne hCcomplete hCconv 0\n set R := Metric.infDist (0 : EuclideanSpace ℝ (Fin n)) (convexHull ℝ S) with hRdef\n have hRnn : 0 ≤ R := by rw [hRdef]; exact Metric.infDist_nonneg\n have hR_iInf : R = ⨅ w : convexHull ℝ S,\n ‖(0 : EuclideanSpace ℝ (Fin n)) - (w : EuclideanSpace ℝ (Fin n))‖ := by\n rw [hRdef, Metric.infDist_eq_iInf]; exact iInf_congr fun w => dist_eq_norm 0 _\n have hpR : ‖p‖ = R := by rw [hR_iInf, ← hpeq]; simp\n have hvar := (norm_eq_iInf_iff_real_inner_le_zero hCconv hpC).mp hpeq\n have hpge : ∀ w ∈ convexHull ℝ S, R ^ 2 ≤ inner ℝ p w := by\n intro w hw\n have h0 := hvar w hw\n have hpp : inner ℝ p p = R ^ 2 := by rw [real_inner_self_eq_norm_sq, hpR]\n have hexp : inner ℝ ((0 : EuclideanSpace ℝ (Fin n)) - p) (w - p)\n = inner ℝ p p - inner ℝ p w := by\n rw [inner_sub_left, inner_sub_right, inner_sub_right, inner_zero_left, inner_zero_left]; ring\n rw [hexp, hpp] at h0; linarith\n have hupper : ∀ a : {a : EuclideanSpace ℝ (Fin n) // ‖a‖ = 1},\n (⨅ y : A, inner ℝ a.val (φ y.val)) ≤ R := by\n intro a\n have hbddb : BddBelow (Set.range (fun y : A => inner ℝ a.val (φ y.val))) := by\n apply Set.Finite.bddBelow\n apply Set.Finite.subset (hSfin.image (fun s => inner ℝ a.val s))\n rintro _ ⟨y, rfl⟩\n exact ⟨φ y.val, hφmem y.val y.2, rfl⟩\n have hlin : IsLinearMap ℝ (fun w : EuclideanSpace ℝ (Fin n) => (inner ℝ a.val w : ℝ)) := by\n refine ⟨fun u v => ?_, fun r v => ?_⟩\n · change inner ℝ a.val (u + v) = inner ℝ a.val u + inner ℝ a.val v\n rw [inner_add_right]\n · change inner ℝ a.val (r • v) = r • inner ℝ a.val v\n rw [real_inner_smul_right, smul_eq_mul]\n have hSc : S ⊆ {w : EuclideanSpace ℝ (Fin n) |\n (⨅ y : A, inner ℝ a.val (φ y.val)) ≤ inner ℝ a.val w} := by\n intro s hs\n rw [hS] at hs\n obtain ⟨y, hyA, hsy⟩ := hs\n have hsφ : s = φ y := by ext i; simpa [hφi] using hsy i\n rw [Set.mem_setOf_eq, hsφ]\n exact ciInf_le hbddb ⟨y, hyA⟩\n have hpH : (⨅ y : A, inner ℝ a.val (φ y.val)) ≤ inner ℝ a.val p :=\n convexHull_min hSc (convex_halfSpace_ge hlin _) hpC\n have hcs : inner ℝ a.val p ≤ R := by\n have h := real_inner_le_norm a.val p\n rw [a.2, one_mul, hpR] at h; exact h\n exact le_trans hpH hcs\n have hbddA : BddAbove (Set.range (fun a : {a : EuclideanSpace ℝ (Fin n) // ‖a‖ = 1} =>\n ⨅ y : A, inner ℝ a.val (φ y.val))) := ⟨R, by rintro z ⟨a, rfl⟩; exact hupper a⟩\n refine ⟨?_, ?_⟩\n · have hLeq : (⨆ a : {a : EuclideanSpace ℝ (Fin n) // ‖a‖ = 1},\n ⨅ y : A, ∑ i, a.val i * (if x i = y.val i then (0 : ℝ) else 1))\n = ⨆ a : {a : EuclideanSpace ℝ (Fin n) // ‖a‖ = 1},\n ⨅ y : A, inner ℝ a.val (φ y.val) :=\n iSup_congr fun a => iInf_congr fun y => hterm a.val y.val\n rw [hLeq]\n rcases Nat.eq_zero_or_pos n with hn0 | hnpos\n · have hemp : IsEmpty {a : EuclideanSpace ℝ (Fin n) // ‖a‖ = 1} := by\n subst hn0\n refine ⟨fun a => ?_⟩\n obtain ⟨a, ha⟩ := a\n have : a = 0 := Subsingleton.elim a 0\n rw [this] at ha; simp at ha\n have hRz : R = 0 := by\n have h0 : (0 : EuclideanSpace ℝ (Fin n)) ∈ convexHull ℝ S := by\n obtain ⟨q, hq⟩ := hCne\n have : q = 0 := by subst hn0; exact Subsingleton.elim q 0\n rwa [this] at hq\n rw [hRdef]; exact Metric.infDist_zero_of_mem h0\n rw [hRz]\n haveI := hemp\n simp [Real.iSup_of_isEmpty]\n · haveI hsph : Nonempty {a : EuclideanSpace ℝ (Fin n) // ‖a‖ = 1} :=\n ⟨⟨EuclideanSpace.single (⟨0, hnpos⟩ : Fin n) (1 : ℝ), by\n rw [EuclideanSpace.norm_single]; norm_num⟩⟩\n apply le_antisymm\n · exact ciSup_le hupper\n · rcases eq_or_lt_of_le hRnn with hR0 | hRpos\n · rw [← hR0]\n have hwit : (0 : ℝ) ≤ ⨅ y : A,\n inner ℝ (EuclideanSpace.single (⟨0, hnpos⟩ : Fin n) (1 : ℝ)) (φ y.val) := by\n apply le_ciInf\n intro y\n rw [inner_eq_sum']\n apply Finset.sum_nonneg\n intro i _\n apply mul_nonneg\n · rw [EuclideanSpace.single_apply]; split <;> norm_num\n · rw [hφi]; split <;> norm_num\n calc (0 : ℝ)\n ≤ ⨅ y : A, inner ℝ (EuclideanSpace.single (⟨0, hnpos⟩ : Fin n) (1 : ℝ)) (φ y.val) :=\n hwit\n _ ≤ ⨆ a : {a : EuclideanSpace ℝ (Fin n) // ‖a‖ = 1},\n ⨅ y : A, inner ℝ a.val (φ y.val) :=\n le_ciSup hbddA ⟨EuclideanSpace.single (⟨0, hnpos⟩ : Fin n) (1 : ℝ), by\n rw [EuclideanSpace.norm_single]; norm_num⟩\n · have hunit : ‖(R⁻¹ • p : EuclideanSpace ℝ (Fin n))‖ = 1 := by\n rw [norm_smul, hpR, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hRpos),\n inv_mul_cancel₀ (ne_of_gt hRpos)]\n have hge : R ≤ ⨅ y : A, inner ℝ (R⁻¹ • p) (φ y.val) := by\n apply le_ciInf\n intro y\n have hφyC : φ y.val ∈ convexHull ℝ S := subset_convexHull ℝ S (hφmem y.val y.2)\n have h1 : R ^ 2 ≤ inner ℝ p (φ y.val) := hpge _ hφyC\n rw [real_inner_smul_left]\n calc R = R⁻¹ * R ^ 2 := by\n rw [pow_two, ← mul_assoc, inv_mul_cancel₀ (ne_of_gt hRpos), one_mul]\n _ ≤ R⁻¹ * inner ℝ p (φ y.val) :=\n mul_le_mul_of_nonneg_left h1 (le_of_lt (inv_pos.mpr hRpos))\n calc R ≤ ⨅ y : A, inner ℝ (R⁻¹ • p) (φ y.val) := hge\n _ ≤ ⨆ a : {a : EuclideanSpace ℝ (Fin n) // ‖a‖ = 1},\n ⨅ y : A, inner ℝ a.val (φ y.val) :=\n le_ciSup hbddA ⟨R⁻¹ • p, hunit⟩\n · rw [hRdef, Metric.infDist_eq_iInf]\n exact iInf_congr fun v => dist_zero_left _", "time_seconds": 1444, "topic": "Random walks & large deviations" }, { "question_id": 400, "problem_name": "stochastic_processes_poisson_age_conditional", "informal_statement": "Let (N(t)) be a Poisson process of rate λ with residual life Y(t) and age Z(t) at time t.\n\nShow that Pr[Y(t) > x | Z(t) = s] = e^{−λx}, i.e. the residual life is exponential of rate λ and independent of the current age.\n\nMeasure age from the last arrival, using the origin if there has been no arrival: 0≤Z(t)≤t and Z(0)=0. Tail identities at deterministic times hold almost surely, allowing the null event of an arrival exactly at a boundary.", "lean_statement": "theorem q400_poisson_age_conditional (μ : Measure Ω) [IsProbabilityMeasure μ]\n (lam : ℝ≥0) (hlam : 0 < lam) (t : ℝ≥0)\n (N : ℝ≥0 → Ω → ℕ) (hN : IsPoissonProcess μ lam N)\n (Y Z : Ω → ℝ) (hYmeas : Measurable Y) (hZmeas : Measurable Z)\n (hYdef : ∀ y : ℝ≥0, ∀ᵐ ω ∂μ,\n ((y : ℝ) < Y ω ↔ N (t + y) ω - N t ω = 0))\n (hZrange : ∀ᵐ ω ∂μ, Z ω ∈ Set.Icc (0 : ℝ) (t : ℝ))\n (hZdef : ∀ s : ℝ≥0, s < t → ∀ᵐ ω ∂μ,\n ((s : ℝ) < Z ω ↔ N t ω - N (t - s) ω = 0)) :\n HasLaw Y (expMeasure (lam : ℝ)) μ ∧ IndepFun Y Z μ := by\n sorry\n", "true_to_source": "abstract", "proof": "theorem q400_poisson_age_conditional (μ : Measure Ω) [IsProbabilityMeasure μ]\n (lam : ℝ≥0) (hlam : 0 < lam) (t : ℝ≥0)\n (N : ℝ≥0 → Ω → ℕ) (hN : IsPoissonProcess μ lam N)\n (Y Z : Ω → ℝ) (hYmeas : Measurable Y) (hZmeas : Measurable Z)\n (hYdef : ∀ y : ℝ≥0, ∀ᵐ ω ∂μ,\n ((y : ℝ) < Y ω ↔ N (t + y) ω - N t ω = 0))\n (hZrange : ∀ᵐ ω ∂μ, Z ω ∈ Set.Icc (0 : ℝ) (t : ℝ))\n (hZdef : ∀ s : ℝ≥0, s < t → ∀ᵐ ω ∂μ,\n ((s : ℝ) < Z ω ↔ N t ω - N (t - s) ω = 0)) :\n HasLaw Y (expMeasure (lam : ℝ)) μ ∧ IndepFun Y Z μ := by\n have hlamR : (0 : ℝ) < (lam : ℝ) := by exact_mod_cast hlam\n have hYset : ∀ a : ℝ≥0, {ω | (a : ℝ) < Y ω} =ᵐ[μ] {ω | N (t + a) ω - N t ω = 0} := by\n intro a\n rw [Filter.eventuallyEq_set]\n filter_upwards [hYdef a] with ω h\n simpa using h\n have hZset : ∀ b : ℝ≥0, b < t →\n {ω | (b : ℝ) < Z ω} =ᵐ[μ] {ω | N t ω - N (t - b) ω = 0} := by\n intro b hb\n rw [Filter.eventuallyEq_set]\n filter_upwards [hZdef b hb] with ω h\n simpa using h\n have hYpos : ∀ᵐ ω ∂μ, (0 : ℝ) < Y ω := by\n filter_upwards [hYdef 0] with ω h\n have h0 : ((0 : ℝ≥0) : ℝ) < Y ω := h.mpr (by simp)\n simpa using h0\n have hZnn : ∀ᵐ ω ∂μ, (0 : ℝ) ≤ Z ω := by\n filter_upwards [hZrange] with ω h using h.1\n have hZle : ∀ᵐ ω ∂μ, Z ω ≤ (t : ℝ) := by\n filter_upwards [hZrange] with ω h using h.2\n have hYval : ∀ a : ℝ≥0, μ.real {ω | (a : ℝ) < Y ω} = Real.exp (-((lam : ℝ) * (a : ℝ))) := by\n intro a\n have hval : μ.real {ω | N (t + a) ω - N t ω = 0} = (poissonMeasure (lam * a)).real {0} := by\n have hlaw := hN.2.2 t (t + a) le_self_add\n rw [add_tsub_cancel_left] at hlaw\n exact hlaw.measureReal_eq (p := fun n : ℕ => n = 0) (measurableSet_singleton 0)\n rw [measureReal_congr (hYset a), hval, poissonMeasure_real_singleton]\n simp [NNReal.coe_mul]\n refine ⟨⟨hYmeas.aemeasurable, ?_⟩, ?_⟩\n · haveI : IsProbabilityMeasure (μ.map Y) := Measure.isProbabilityMeasure_map hYmeas.aemeasurable\n haveI : IsProbabilityMeasure (expMeasure (lam : ℝ)) := isProbabilityMeasure_expMeasure hlamR\n apply Measure.eq_of_cdf\n apply StieltjesFunction.ext\n intro x\n rw [cdf_expMeasure_eq hlamR x, cdf_eq_real, Measure.real,\n Measure.map_apply hYmeas measurableSet_Iic, ← Measure.real]\n have hcompl : Y ⁻¹' Set.Iic x = {ω | x < Y ω}ᶜ := by ext ω; simp [not_lt]\n have hmeas : MeasurableSet {ω | x < Y ω} := hYmeas measurableSet_Ioi\n rw [hcompl, measureReal_compl hmeas, probReal_univ]\n by_cases hx : 0 ≤ x\n · rw [if_pos hx]\n have hv := hYval x.toNNReal\n rw [Real.coe_toNNReal x hx] at hv\n rw [hv]\n · rw [if_neg hx]\n have hU : {ω | x < Y ω} =ᵐ[μ] (Set.univ : Set Ω) := by\n rw [Filter.eventuallyEq_set]\n filter_upwards [hYpos] with ω h\n simp only [Set.mem_setOf_eq, Set.mem_univ, iff_true]\n exact lt_trans (not_le.mp hx) h\n rw [measureReal_congr hU, probReal_univ]; norm_num\n · rw [IndepFun_iff_Indep Y Z μ]\n have hcY : MeasurableSpace.comap Y (inferInstance : MeasurableSpace ℝ)\n = MeasurableSpace.generateFrom (Set.range fun a : ℝ => Y ⁻¹' Set.Ioi a) := by\n have hb : (inferInstance : MeasurableSpace ℝ) = borel ℝ := BorelSpace.measurable_eq\n rw [hb, borel_eq_generateFrom_Ioi ℝ, MeasurableSpace.comap_generateFrom, ← Set.range_comp]\n rfl\n have hcZ : MeasurableSpace.comap Z (inferInstance : MeasurableSpace ℝ)\n = MeasurableSpace.generateFrom (Set.range fun b : ℝ => Z ⁻¹' Set.Ioi b) := by\n have hb : (inferInstance : MeasurableSpace ℝ) = borel ℝ := BorelSpace.measurable_eq\n rw [hb, borel_eq_generateFrom_Ioi ℝ, MeasurableSpace.comap_generateFrom, ← Set.range_comp]\n rfl\n have hp1pi : IsPiSystem (Set.range fun a : ℝ => Y ⁻¹' Set.Ioi a) := by\n rintro _ ⟨a, rfl⟩ _ ⟨b, rfl⟩ _\n exact ⟨max a b, by rw [← Set.preimage_inter, Set.Ioi_inter_Ioi]⟩\n have hp2pi : IsPiSystem (Set.range fun b : ℝ => Z ⁻¹' Set.Ioi b) := by\n rintro _ ⟨a, rfl⟩ _ ⟨b, rfl⟩ _\n exact ⟨max a b, by rw [← Set.preimage_inter, Set.Ioi_inter_Ioi]⟩\n have hIndep : IndepSets (Set.range fun a : ℝ => Y ⁻¹' Set.Ioi a)\n (Set.range fun b : ℝ => Z ⁻¹' Set.Ioi b) μ := by\n rw [IndepSets_iff]\n rintro _ _ ⟨a, rfl⟩ ⟨b, rfl⟩\n show μ ({ω | a < Y ω} ∩ {ω | b < Z ω}) = μ {ω | a < Y ω} * μ {ω | b < Z ω}\n by_cases ha : 0 ≤ a\n · by_cases hb0 : 0 ≤ b\n · by_cases hbt : b < (t : ℝ)\n · have hca : ((a.toNNReal : ℝ)) = a := Real.coe_toNNReal a ha\n have hcb : ((b.toNNReal : ℝ)) = b := Real.coe_toNNReal b hb0\n have hb't : b.toNNReal < t := by rw [← NNReal.coe_lt_coe, hcb]; exact hbt\n have hSeq : {ω | a < Y ω} =ᵐ[μ] {ω | N (t + a.toNNReal) ω - N t ω = 0} := by\n have h := hYset a.toNNReal; rw [hca] at h; exact h\n have hTeq : {ω | b < Z ω} =ᵐ[μ] {ω | N t ω - N (t - b.toNNReal) ω = 0} := by\n have h := hZset b.toNNReal hb't; rw [hcb] at h; exact h\n have hindep : IndepFun (fun ω => N (t + a.toNNReal) ω - N t ω)\n (fun ω => N t ω - N (t - b.toNNReal) ω) μ :=\n (hN.2.1.indepFun_sub_sub (r := t - b.toNNReal) (s := t) (t := t + a.toNNReal)\n tsub_le_self le_self_add).symm\n have hmul := hindep.measure_inter_preimage_eq_mul {0} {0}\n (measurableSet_singleton 0) (measurableSet_singleton 0)\n rw [measure_congr (hSeq.inter hTeq), measure_congr hSeq, measure_congr hTeq]\n exact hmul\n · push_neg at hbt\n have hE : {ω | b < Z ω} =ᵐ[μ] (∅ : Set Ω) := by\n rw [Filter.eventuallyEq_set]\n filter_upwards [hZle] with ω h\n simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false, not_lt]\n linarith\n have hz : μ ({ω | a < Y ω} ∩ {ω | b < Z ω}) = 0 := by\n rw [measure_congr ((ae_eq_refl _).inter hE), Set.inter_empty, measure_empty]\n rw [hz, measure_congr hE, measure_empty, mul_zero]\n · push_neg at hb0\n have hU : {ω | b < Z ω} =ᵐ[μ] (Set.univ : Set Ω) := by\n rw [Filter.eventuallyEq_set]\n filter_upwards [hZnn] with ω h\n simp only [Set.mem_setOf_eq, Set.mem_univ, iff_true]\n linarith\n rw [measure_congr ((ae_eq_refl _).inter hU), Set.inter_univ,\n measure_congr hU, measure_univ, mul_one]\n · push_neg at ha\n have hU : {ω | a < Y ω} =ᵐ[μ] (Set.univ : Set Ω) := by\n rw [Filter.eventuallyEq_set]\n filter_upwards [hYpos] with ω h\n simp only [Set.mem_setOf_eq, Set.mem_univ, iff_true]\n linarith\n rw [measure_congr (hU.inter (ae_eq_refl _)), Set.univ_inter,\n measure_congr hU, measure_univ, one_mul]\n exact IndepSets.indep (measurable_iff_comap_le.mp hYmeas)\n (measurable_iff_comap_le.mp hZmeas) hp1pi hp2pi hcY hcZ hIndep", "time_seconds": 1765, "topic": "Poisson processes" }, { "question_id": 430, "problem_name": "stochastic_processes_poisson_superposition_2", "informal_statement": "Let (N^1(t)) and (N^2(t)) be independent Poisson processes of rates λ_1 and λ_2.\n\nShow that the superposition N(t) = N^1(t) + N^2(t) is a Poisson process of rate λ_1 + λ_2.", "lean_statement": "theorem q430_poisson_superposition (μ : Measure Ω) [IsProbabilityMeasure μ]\n (lam1 lam2 : ℝ≥0) (N1 N2 : ℝ≥0 → Ω → ℕ)\n (hN1 : IsPoissonProcess μ lam1 N1) (hN2 : IsPoissonProcess μ lam2 N2)\n (hindep : IndepFun (fun ω => (fun t => N1 t ω)) (fun ω => (fun t => N2 t ω)) μ) :\n IsPoissonProcess μ (lam1 + lam2) (fun t ω => N1 t ω + N2 t ω) := by\n sorry\n", "true_to_source": "abstract", "proof": "private lemma indepFun_to_iIndepFun_fin2 {Ω : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}\n [IsProbabilityMeasure μ] {α : Type*} [MeasurableSpace α] {f g : Ω → α}\n (h : IndepFun f g μ) : iIndepFun ![f, g] μ := by\n rw [iIndepFun_iff_measure_inter_preimage_eq_mul]\n intro s sets hsets\n fin_cases s <;> simp_all [IndepFun.measure_inter_preimage_eq_mul]\n\nprivate lemma group_curry {Ω : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}\n [IsProbabilityMeasure μ] {n : ℕ} (X : Fin 2 → Fin n → Ω → ℕ)\n (hXm : ∀ i j, Measurable (X i j))\n (hpair : iIndepFun (fun (q : Fin n × Fin 2) ω => X q.2 q.1 ω) μ) :\n iIndepFun (fun (j : Fin n) ω (i : Fin 2) => X i j ω) μ := by\n haveI hpm : ∀ (i : Fin 2) (j : Fin n),\n IsProbabilityMeasure (μ.map (fun ω => X i j ω)) :=\n fun i j => Measure.isProbabilityMeasure_map (hXm i j).aemeasurable\n have hfiber : ∀ j : Fin n, μ.map (fun ω (i : Fin 2) => X i j ω)\n = Measure.infinitePi (fun i => μ.map (fun ω => X i j ω)) := by\n intro j\n have hj : iIndepFun (fun (i : Fin 2) ω => X i j ω) μ :=\n hpair.precomp (g := fun i : Fin 2 => (j, i)) (fun a b h => by simpa using h)\n exact hj.map_fun_eq_infinitePi_map (fun i => hXm i j)\n rw [iIndepFun_iff_map_fun_eq_infinitePi_map (fun j => by fun_prop)]\n simp_rw [hfiber]\n have hcurry : (fun ω (j : Fin n) (i : Fin 2) => X i j ω)\n = (MeasurableEquiv.curry (Fin n) (Fin 2) ℕ) ∘ (fun ω (q : Fin n × Fin 2) => X q.2 q.1 ω) :=\n rfl\n rw [hcurry, ← Measure.map_map (by fun_prop) (by fun_prop),\n ← Measure.infinitePi_map_curry (fun (j : Fin n) (i : Fin 2) => μ.map (fun ω => X i j ω))]\n congr 1\n exact hpair.map_fun_eq_infinitePi_map (fun q => hXm q.2 q.1)\n\nprivate lemma group_two {Ω : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}\n [IsProbabilityMeasure μ] {n : ℕ} (A B : Fin n → Ω → ℕ)\n (hAm : ∀ j, Measurable (A j)) (hBm : ∀ j, Measurable (B j))\n (hAi : iIndepFun A μ) (hBi : iIndepFun B μ)\n (hcross : IndepFun (fun ω j => A j ω) (fun ω j => B j ω) μ) :\n iIndepFun (fun (j : Fin n) ω => A j ω + B j ω) μ := by\n have hXm : ∀ (i : Fin 2) (j : Fin n),\n Measurable ((![A, B] : Fin 2 → Fin n → Ω → ℕ) i j) := by\n intro i j; fin_cases i\n · exact hAm j\n · exact hBm j\n have hpair : iIndepFun\n (fun (q : Fin n × Fin 2) ω => (![A, B] : Fin 2 → Fin n → Ω → ℕ) q.2 q.1 ω) μ := by\n have h2 : ∀ i : Fin 2, iIndepFun ((![A, B] : Fin 2 → Fin n → Ω → ℕ) i) μ := by\n intro i; fin_cases i\n · exact hAi\n · exact hBi\n have h1 : iIndepFun\n (fun (i : Fin 2) ω (x : Fin n) => (![A, B] : Fin 2 → Fin n → Ω → ℕ) i x ω) μ := by\n have he : (fun (i : Fin 2) ω (x : Fin n) => (![A, B] : Fin 2 → Fin n → Ω → ℕ) i x ω)\n = ![fun ω j => A j ω, fun ω j => B j ω] := by\n funext i; fin_cases i <;> rfl\n rw [he]; exact indepFun_to_iIndepFun_fin2 hcross\n exact (iIndepFun_uncurry' hXm h1 h2).precomp Prod.swap_injective\n have hgroup := group_curry (![A, B] : Fin 2 → Fin n → Ω → ℕ) hXm hpair\n have hcomp := hgroup.comp (fun _ : Fin n => fun v : Fin 2 → ℕ => v 0 + v 1)\n (fun _ => by fun_prop)\n have heq : (fun (j : Fin n) ω => A j ω + B j ω)\n = fun (j : Fin n) => (fun v : Fin 2 → ℕ => v 0 + v 1)\n ∘ (fun ω (i : Fin 2) => (![A, B] : Fin 2 → Fin n → Ω → ℕ) i j ω) := by\n funext j ω; simp [Function.comp]\n rw [heq]; exact hcomp\n\ntheorem q430_poisson_superposition (μ : Measure Ω) [IsProbabilityMeasure μ]\n (lam1 lam2 : ℝ≥0) (N1 N2 : ℝ≥0 → Ω → ℕ)\n (hN1 : IsPoissonProcess μ lam1 N1) (hN2 : IsPoissonProcess μ lam2 N2)\n (hindep : IndepFun (fun ω => (fun t => N1 t ω)) (fun ω => (fun t => N2 t ω)) μ) :\n IsPoissonProcess μ (lam1 + lam2) (fun t ω => N1 t ω + N2 t ω) := by\n obtain ⟨⟨hN1meas, hN1zero, hN1path⟩, hN1inc, hN1law⟩ := hN1\n obtain ⟨⟨hN2meas, hN2zero, hN2path⟩, hN2inc, hN2law⟩ := hN2\n refine ⟨⟨?_, ?_, ?_⟩, ?_, ?_⟩\n · -- measurability\n exact fun t => (hN1meas t).add (hN2meas t)\n · -- starts at zero\n filter_upwards [hN1zero, hN2zero] with ω h1 h2\n simp [h1, h2]\n · -- monotone + right-continuous paths\n filter_upwards [hN1path, hN2path] with ω hp1 hp2\n exact ⟨hp1.1.add hp2.1, fun t => (hp1.2 t).add (hp2.2 t)⟩\n · -- independent increments\n intro n t ht\n set A : Fin n → Ω → ℕ := fun i ω => N1 (t i.succ) ω - N1 (t i.castSucc) ω with hAdef\n set B : Fin n → Ω → ℕ := fun i ω => N2 (t i.succ) ω - N2 (t i.castSucc) ω with hBdef\n have hAi : iIndepFun A μ := hN1inc n t ht\n have hBi : iIndepFun B μ := hN2inc n t ht\n have hAm : ∀ i, Measurable (A i) := fun i => (hN1meas _).sub (hN1meas _)\n have hBm : ∀ i, Measurable (B i) := fun i => (hN2meas _).sub (hN2meas _)\n have hcross : IndepFun (fun ω j => A j ω) (fun ω j => B j ω) μ := by\n have hφ : Measurable\n (fun (p : ℝ≥0 → ℕ) (j : Fin n) => p (t j.succ) - p (t j.castSucc)) := by fun_prop\n exact hindep.comp hφ hφ\n have hclean := group_two A B hAm hBm hAi hBi hcross\n refine iIndepFun.congr ?_ hclean\n intro i\n filter_upwards [hN1path, hN2path] with ω hp1 hp2\n have h1 : N1 (t i.castSucc) ω ≤ N1 (t i.succ) ω := hp1.1 (ht (Fin.castSucc_le_succ i))\n have h2 : N2 (t i.castSucc) ω ≤ N2 (t i.succ) ω := hp2.1 (ht (Fin.castSucc_le_succ i))\n simp only [hAdef, hBdef]\n omega\n · -- Poisson increment law\n intro a b hab\n have hA := hN1law a b hab\n have hB := hN2law a b hab\n have hcross : IndepFun (fun ω => N1 b ω - N1 a ω) (fun ω => N2 b ω - N2 a ω) μ := by\n have hφ : Measurable (fun p : ℝ≥0 → ℕ => p b - p a) := by fun_prop\n exact hindep.comp hφ hφ\n have hsum := hcross.hasLaw_add_poissonMeasure hA hB\n rw [show (lam1 + lam2) * (b - a) = lam1 * (b - a) + lam2 * (b - a) from by ring]\n refine hsum.congr ?_\n filter_upwards [hN1path, hN2path] with ω hp1 hp2\n have h1 : N1 a ω ≤ N1 b ω := hp1.1 hab\n have h2 : N2 a ω ≤ N2 b ω := hp2.1 hab\n simp only [Pi.add_apply]\n omega", "time_seconds": 1429, "topic": "Poisson processes" } ]