| import Mathlib |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped Topology ENNReal NNReal |
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| namespace Problem367 |
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| variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
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| theorem ae_tendsto_zero_of_tail_tsum_ne_top (X : ℕ → Ω → ℝ) |
| (hX : ∀ ε : ℝ, 0 < ε → (∑' n, μ {ω | ε ≤ |X n ω|}) ≠ ∞) : |
| ∀ᵐ ω ∂μ, Tendsto (fun n => X n ω) atTop (𝓝 0) := by |
| have hsmall : ∀ k : ℕ, ∀ᵐ ω ∂μ, |
| ∀ᶠ n in atTop, |X n ω| < 1 / (k + 1 : ℝ) := by |
| intro k |
| have h := ae_eventually_notMem (hX (1 / (k + 1 : ℝ)) (by positivity)) |
| simpa only [Set.mem_ofPred_eq, not_le] using h |
| filter_upwards [ae_all_iff.mpr hsmall] with ω hω |
| rw [Metric.tendsto_atTop] |
| intro ε hε |
| obtain ⟨k, hk⟩ := exists_nat_one_div_lt hε |
| obtain ⟨N, hN⟩ := eventually_atTop.mp (hω k) |
| exact ⟨N, fun n hn => by simpa [Real.dist_eq] using (hN n hn).trans hk⟩ |
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| theorem ae_tendsto_zero_of_summable_tail [IsFiniteMeasure μ] (X : ℕ → Ω → ℝ) |
| (hX : ∀ ε : ℝ, 0 < ε → Summable (fun n => μ.real {ω | ε ≤ |X n ω|})) : |
| ∀ᵐ ω ∂μ, Tendsto (fun n => X n ω) atTop (𝓝 0) := by |
| apply ae_tendsto_zero_of_tail_tsum_ne_top X |
| intro ε hε |
| have h := (hX ε hε).tsum_ofReal_ne_top |
| simpa only [Measure.real, ENNReal.ofReal_toReal (measure_ne_top _ _)] using h |
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| theorem ae_tendsto_zero_of_exp_tail [IsFiniteMeasure μ] (X : ℕ → Ω → ℝ) |
| (hX : ∀ ε : ℝ, 0 < ε → ∃ C c : ℝ, 0 < c ∧ |
| ∀ n, μ.real {ω | ε ≤ |X n ω|} ≤ C * Real.exp (-(n : ℝ) * c)) : |
| ∀ᵐ ω ∂μ, Tendsto (fun n => X n ω) atTop (𝓝 0) := by |
| apply ae_tendsto_zero_of_summable_tail X |
| intro ε hε |
| obtain ⟨C, c, hc, hbound⟩ := hX ε hε |
| have hsum : Summable (fun n : ℕ => C * Real.exp (-(n : ℝ) * c)) := by |
| simpa only [mul_neg, neg_mul] using |
| (Real.summable_exp_nat_mul_iff.mpr (neg_neg_of_pos hc)).mul_left C |
| exact hsum.of_nonneg_of_le (fun _ => measureReal_nonneg) hbound |
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| theorem ae_all_tendsto_zero_of_exp_tail {ι : Type*} [Countable ι] [IsFiniteMeasure μ] |
| (X : ι → ℕ → Ω → ℝ) |
| (hX : ∀ i ε, 0 < ε → ∃ C c : ℝ, 0 < c ∧ |
| ∀ n, μ.real {ω | ε ≤ |X i n ω|} ≤ C * Real.exp (-(n : ℝ) * c)) : |
| ∀ᵐ ω ∂μ, ∀ i, Tendsto (fun n => X i n ω) atTop (𝓝 0) := |
| ae_all_iff.mpr (fun i => ae_tendsto_zero_of_exp_tail (X i) (hX i)) |
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| theorem subgaussian_abs_tail [IsFiniteMeasure μ] {X : Ω → ℝ} {c : ℝ≥0} |
| (hX : HasSubgaussianMGF X c μ) {ε : ℝ} (hε : 0 ≤ ε) : |
| μ.real {ω | ε ≤ |X ω|} ≤ 2 * Real.exp (-ε ^ 2 / (2 * c)) := by |
| have hset : {ω | ε ≤ |X ω|} = {ω | ε ≤ X ω} ∪ {ω | ε ≤ -X ω} := by |
| ext ω |
| simpa only [Set.mem_ofPred_eq, Set.mem_union] using (le_abs (a := ε) (b := X ω)) |
| rw [hset] |
| calc |
| μ.real ({ω | ε ≤ X ω} ∪ {ω | ε ≤ -X ω}) ≤ |
| μ.real {ω | ε ≤ X ω} + μ.real {ω | ε ≤ -X ω} := measureReal_union_le _ _ |
| _ ≤ Real.exp (-ε ^ 2 / (2 * c)) + Real.exp (-ε ^ 2 / (2 * c)) := |
| add_le_add (hX.measure_ge_le hε) (hX.neg.measure_ge_le hε) |
| _ = 2 * Real.exp (-ε ^ 2 / (2 * c)) := by ring |
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| theorem ae_tendsto_div_succ_zero_of_subgaussian [IsFiniteMeasure μ] |
| (X : ℕ → Ω → ℝ) |
| (hX : ∀ n, HasSubgaussianMGF (X n) ((n : ℝ≥0) + 1) μ) : |
| ∀ᵐ ω ∂μ, Tendsto (fun n => X n ω / ((n : ℝ) + 1)) atTop (𝓝 0) := by |
| apply ae_tendsto_zero_of_exp_tail |
| intro ε hε |
| refine ⟨2, ε ^ 2 / 2, by positivity, ?_⟩ |
| intro n |
| have hn : 0 < (n : ℝ) + 1 := by positivity |
| have hset : {ω | ε ≤ |X n ω / ((n : ℝ) + 1)|} = |
| {ω | ε * ((n : ℝ) + 1) ≤ |X n ω|} := by |
| ext ω |
| simp only [Set.mem_ofPred_eq, abs_div, abs_of_pos hn, le_div_iff₀ hn] |
| rw [hset] |
| have ht := subgaussian_abs_tail (hX n) (mul_nonneg hε.le hn.le) |
| simp only [NNReal.coe_add, NNReal.coe_natCast, NNReal.coe_one] at ht |
| refine ht.trans ?_ |
| have heq : -(ε * ((n : ℝ) + 1)) ^ 2 / (2 * ((n : ℝ) + 1)) = |
| -((n : ℝ) + 1) * (ε ^ 2 / 2) := by |
| field_simp [ne_of_gt hn] |
| rw [heq] |
| gcongr |
| nlinarith [sq_nonneg ε] |
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| theorem hasSubgaussianMGF_of_hasLaw_gaussian {X : Ω → ℝ} {v : ℝ≥0} |
| (hX : HasLaw X (gaussianReal 0 v) μ) : HasSubgaussianMGF X v μ := by |
| have hg : HasSubgaussianMGF id v (gaussianReal 0 v) := by |
| constructor |
| · intro t |
| exact integrable_exp_mul_gaussianReal t |
| · intro t |
| simp [mgf_id_gaussianReal] |
| rw [← hX.map_eq] at hg |
| simpa only [Function.id_comp] using hg.of_map hX.aemeasurable |
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| theorem ae_brownian_div_succ_tendsto_zero (B : ℝ≥0 → Ω → ℝ) |
| (hB : IsPreBrownianReal B μ) : |
| ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => B ((n : ℝ≥0) + 1) ω / ((n : ℝ) + 1)) |
| atTop (𝓝 0) := by |
| let : IsProbabilityMeasure μ := (hB.hasLaw_eval 0).isProbabilityMeasure |
| exact ae_tendsto_div_succ_zero_of_subgaussian |
| (fun n => B ((n : ℝ≥0) + 1)) |
| (fun n => hasSubgaussianMGF_of_hasLaw_gaussian (hB.hasLaw_eval _)) |
|
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| theorem ae_brownian_div_nat_tendsto_zero (B : ℝ≥0 → Ω → ℝ) |
| (hB : IsPreBrownianReal B μ) : |
| ∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => B n ω / (n : ℝ)) atTop (𝓝 0) := by |
| filter_upwards [ae_brownian_div_succ_tendsto_zero B hB] with ω hω |
| apply (tendsto_add_atTop_iff_nat 1).mp |
| simpa only [Nat.cast_add, Nat.cast_one] using hω |
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| theorem ae_all_brownian_div_nat_tendsto_zero {ι : Type*} [Countable ι] |
| (B : ι → ℝ≥0 → Ω → ℝ) (hB : ∀ i, IsPreBrownianReal (B i) μ) : |
| ∀ᵐ ω ∂μ, ∀ i, Tendsto (fun n : ℕ => B i n ω / (n : ℝ)) atTop (𝓝 0) := |
| ae_all_iff.mpr (fun i => ae_brownian_div_nat_tendsto_zero (B i) (hB i)) |
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| theorem ae_brownian_signal_recovery (B : ℝ≥0 → Ω → ℝ) |
| (hB : IsPreBrownianReal B μ) (signal : Ω → ℝ) : |
| ∀ᵐ ω ∂μ, Tendsto |
| (fun n : ℕ => ((n : ℝ) * signal ω + B n ω) / (n : ℝ)) |
| atTop (𝓝 (signal ω)) := by |
| filter_upwards [ae_brownian_div_nat_tendsto_zero B hB] with ω hω |
| have ht : Tendsto (fun n : ℕ => signal ω + B n ω / (n : ℝ)) |
| atTop (𝓝 (signal ω)) := by |
| simpa only [add_zero] using tendsto_const_nhds.add hω |
| refine ht.congr' ?_ |
| filter_upwards [eventually_ge_atTop 1] with n hn |
| have hn0 : (n : ℝ) ≠ 0 := by exact_mod_cast (by omega : n ≠ 0) |
| field_simp |
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| end Problem367 |
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| end OAI |
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