| import OAI.Probability.Renewal.Decay |
| import OAI.Probability.Renewal.TiltMoments |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology ENNReal |
| namespace HoneycombLocal |
|
|
| lemma integrable_stretched_exp {a c : ℝ} (ha : 0 < a) (hc : 0 < c) : |
| Integrable (fun t : ℝ => Real.exp (-c * |t| ^ a)) := by |
| have hi : IntegrableOn (fun t : ℝ => Real.exp (-c * t ^ a)) (Ioi 0) := by |
| simpa using integrableOn_rpow_mul_exp_neg_mul_rpow (s := 0) (by norm_num) ha hc |
| have hp : IntegrableOn (fun t : ℝ => Real.exp (-c * |t| ^ a)) (Ioi 0) := |
| (integrableOn_congr_fun (fun t ht => by rw [abs_of_pos ht]) measurableSet_Ioi).mpr hi |
| have hn : IntegrableOn (fun t : ℝ => Real.exp (-c * |t| ^ a)) (Iic 0) := by |
| rw [← Measure.map_neg_eq_self (volume : Measure ℝ)] |
| let em : MeasurableEmbedding (fun x : ℝ => -x) := (Homeomorph.neg ℝ).measurableEmbedding |
| rw [em.integrableOn_map_iff] |
| simp only [Function.comp_def, abs_neg, neg_preimage, neg_Iic, neg_zero] |
| exact Iff.mpr integrableOn_Ici_iff_integrableOn_Ioi hp |
| simpa only [Iic_union_Ioi, integrableOn_univ] using hn.union hp |
|
|
| lemma exists_integral_tail_small {g : ℝ → ℝ} (hg : Integrable g) {ε : ℝ} (hε : 0 < ε) : |
| ∃ ω : ℝ, 1 ≤ ω ∧ (∫ t : ℝ in (Icc (-ω) ω)ᶜ, g t) ≤ ε := by |
| have hlim := intervalIntegral_tendsto_integral hg tendsto_neg_atTop_atBot (tendsto_id : Tendsto (fun x : ℝ => x) atTop atTop) |
| have ht : Tendsto (fun ω : ℝ => (∫ t, g t) - ∫ t in -ω..ω, g t) atTop (𝓝 0) := by |
| simpa using (tendsto_const_nhds (x := ∫ t, g t)).sub hlim |
| have hεv := ht.eventually (Iio_mem_nhds hε) |
| obtain ⟨ω, hω, he⟩ := (eventually_ge_atTop (1 : ℝ)).and hεv |>.exists |
| refine ⟨ω, hω, ?_⟩ |
| rw [setIntegral_compl measurableSet_Icc hg, integral_Icc_eq_integral_Ioc, |
| ← intervalIntegral.integral_of_le (by linarith : -ω ≤ ω)] |
| exact he.le |
|
|
| lemma sigma_rpow_bound {a u σ V : ℝ} (ha : 0 < a ∧ a < 1) (hu : 0 < u) |
| (hσ : 0 < σ) (hV : 0 < V) {k : ℕ} (hku : 1 ≤ (k : ℝ) * u ^ a) |
| (hvar : σ ^ 2 ≤ (k : ℝ) * V * u ^ (a - 2)) : |
| σ ^ a ≤ (max V 1) ^ a * (k : ℝ) := by |
| let K := (k : ℝ) * u ^ a |
| let H := max V 1 |
| have hK : 1 ≤ K := hku |
| have hH : 1 ≤ H := le_max_right _ _ |
| have hH0 : 0 < H := hV.trans_le (le_max_left _ _) |
| have hK0 : 0 < K := zero_lt_one.trans_le hK |
| have hid : u ^ (a - 2) * u ^ 2 = u ^ a := by |
| rw [Real.rpow_sub hu, Real.rpow_two] |
| field_simp |
| have hs : (σ * u) ^ 2 ≤ V * K := by |
| have hh := mul_le_mul_of_nonneg_right hvar (sq_nonneg u) |
| simpa only [mul_pow, mul_assoc, hid, K, mul_left_comm V (k : ℝ)] using hh |
| have hHK : 1 ≤ H * K := by nlinarith [mul_nonneg (by linarith : 0 ≤ H - 1) (by linarith : 0 ≤ K - 1)] |
| have hvhk : V * K ≤ H * K := mul_le_mul_of_nonneg_right (le_max_left _ _) hK0.le |
| have hsu : σ * u ≤ H * K := by |
| nlinarith [sq_nonneg (H * K - 1), mul_nonneg (by linarith : 0 ≤ H * K) (by linarith : 0 ≤ H * K - 1)] |
| have hr := Real.rpow_le_rpow (by positivity : 0 ≤ σ * u) hsu ha.1.le |
| rw [Real.mul_rpow hσ.le hu.le, Real.mul_rpow hH0.le hK0.le] at hr |
| have hr' := hr.trans (mul_le_mul_of_nonneg_left (Real.rpow_le_self_of_one_le hK ha.2.le) |
| (Real.rpow_pos_of_pos hH0 a).le) |
| apply (mul_le_mul_iff_left₀ (Real.rpow_pos_of_pos hu a)).mp |
| convert hr' using 1 |
| dsimp [H, K] |
| ring |
|
|
| namespace AnnulusData |
| variable {p : PMF ℕ} {a : ℝ} (D : AnnulusData p a) (M : MomentData p a) |
|
|
| def envelope (t : ℝ) : ℝ := |
| Real.exp (-(D.decayConst / M.varUpper) * t ^ 2) + |
| Real.exp (-(D.decayConst / (max M.varUpper 1) ^ a) * |t| ^ a) |
|
|
| lemma envelope_nonneg (t : ℝ) : 0 ≤ D.envelope M t := by |
| unfold envelope |
| positivity |
|
|
| lemma envelope_integrable (ha : 0 < a) : Integrable (D.envelope M) := by |
| apply Integrable.add |
| · apply integrable_exp_neg_mul_sq |
| exact div_pos D.decayConst_pos M.varUpper_pos |
| · apply integrable_stretched_exp ha |
| exact div_pos D.decayConst_pos (Real.rpow_pos_of_pos (lt_of_lt_of_le zero_lt_one (le_max_right _ _)) a) |
|
|
| lemma normalized_bound (ha : 0 < a ∧ a < 1) {u σ : ℝ} (hu : 0 < u) |
| (huD : u ≤ D.epsilon) (huM : u ≤ M.epsilon) {k : ℕ} |
| (hku : 1 ≤ (k : ℝ) * u ^ a) (hσ : 0 < σ) |
| (hvar : σ ^ 2 = k * natVariance (tilt p u hu.le)) |
| {ρ : ℝ} (hρ : 0 ≤ ρ) |
| (hgap : ∀ t, D.epsilon ≤ |t| → |t| ≤ Real.pi → ‖latticeChar (tilt p u hu.le) t‖ ≤ ρ) |
| (t : ℝ) (ht : |t| ≤ Real.pi * σ) : |
| ‖latticeChar (tilt p u hu.le) (t / σ)‖ ^ k ≤ D.envelope M t + ρ ^ k := by |
| have hV := M.varUpper_pos |
| have hv : σ ^ 2 ≤ (k : ℝ) * M.varUpper * u ^ (a - 2) := by |
| rw [hvar] |
| simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (M.var_upper u hu huM) (Nat.cast_nonneg k) |
| have hσa := sigma_rpow_bound ha hu hσ hV hku hv |
| have hC := D.decayConst_pos |
| have hH : 0 < max M.varUpper 1 := lt_of_lt_of_le zero_lt_one (le_max_right _ _) |
| have hθ : |t / σ| ≤ Real.pi := by |
| rw [abs_div, abs_of_pos hσ, div_le_iff₀ hσ] |
| exact ht |
| by_cases hlo : |t / σ| ≤ u |
| · have hb := pow_le_pow_left₀ (norm_nonneg _) (D.char_bound hu huD hu.le le_rfl hlo) k |
| rw [← Real.exp_nat_mul] at hb |
| have hbase : σ ^ 2 / M.varUpper ≤ (k : ℝ) * u ^ (a - 2) := by |
| apply (div_le_iff₀ hV).mpr |
| nlinarith [hv] |
| have hm := mul_le_mul_of_nonneg_right hbase (by positivity : 0 ≤ D.decayConst * (t / σ) ^ 2) |
| have hid : σ ^ 2 / M.varUpper * (D.decayConst * (t / σ) ^ 2) = |
| D.decayConst / M.varUpper * t ^ 2 := by field_simp |
| rw [hid] at hm |
| have hexp : (k : ℝ) * (-D.decayConst * u ^ (a - 2) * (t / σ) ^ 2) ≤ |
| -(D.decayConst / M.varUpper) * t ^ 2 := by nlinarith |
| have he := hb.trans (Real.exp_le_exp.mpr hexp) |
| exact he.trans (by |
| unfold envelope |
| nlinarith [Real.exp_pos (-(D.decayConst / (max M.varUpper 1) ^ a) * |t| ^ a), pow_nonneg hρ k]) |
| · by_cases hmid : |t / σ| ≤ D.epsilon |
| · have hθ0 : 0 < |t / σ| := hu.trans (lt_of_not_ge hlo) |
| have hb := pow_le_pow_left₀ (norm_nonneg _) (D.intermediate hθ0 hmid hu.le (le_of_not_ge hlo)) k |
| rw [← Real.exp_nat_mul] at hb |
| have hcoef : D.decayConst / (max M.varUpper 1) ^ a ≤ D.decayConst * k / σ ^ a := by |
| apply (div_le_div_iff₀ (Real.rpow_pos_of_pos hH a) (Real.rpow_pos_of_pos hσ a)).mpr |
| nlinarith [mul_le_mul_of_nonneg_left hσa hC.le] |
| have hm := mul_le_mul_of_nonneg_right hcoef (Real.rpow_nonneg (abs_nonneg t) a) |
| have hid : |t / σ| ^ a = |t| ^ a / σ ^ a := by |
| rw [abs_div, abs_of_pos hσ, Real.div_rpow (abs_nonneg t) hσ.le] |
| have hexp : (k : ℝ) * (-D.decayConst * |t / σ| ^ a) ≤ |
| -(D.decayConst / (max M.varUpper 1) ^ a) * |t| ^ a := by |
| rw [hid] |
| simpa only [div_eq_mul_inv, mul_assoc, mul_left_comm, mul_comm, neg_mul, mul_neg] using neg_le_neg hm |
| have he := hb.trans (Real.exp_le_exp.mpr hexp) |
| exact he.trans (by |
| unfold envelope |
| nlinarith [Real.exp_pos (-(D.decayConst / M.varUpper) * t ^ 2), pow_nonneg hρ k]) |
| · have hb := pow_le_pow_left₀ (norm_nonneg _) (hgap (t / σ) (le_of_not_ge hmid) hθ) k |
| exact hb.trans (le_add_of_nonneg_left (D.envelope_nonneg M t)) |
|
|
| end AnnulusData |
| end HoneycombLocal |
|
|
| end |
|
|
| end OAI |
|
|
