| import OAI.Probability.FreeIsing.Estimates.ContinuousCoefficientModulus |
| import OAI.Probability.FreeIsing.Martingales.BracketIsometry |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory Filter |
| open scoped Topology |
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| namespace Problem367.MartingaleCoefficientApproximation |
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| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} |
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| theorem tendsto_integral_sq_coefficient_error [IsFiniteMeasure μ] |
| {I : Type*} [Preorder I] [PseudoMetricSpace I] |
| {F : Filtration I mΩ} {M A H : I → Ω → ℝ} |
| (hM : Martingale M F μ) |
| (hQ : Martingale (fun t ω => M t ω ^ 2 - A t ω) F μ) |
| (hM2 : ∀ t, MemLp (M t) 2 μ) (hH : StronglyAdapted F H) |
| {S : Set I} (hS : IsCompact S) |
| (hcH : ∀ᵐ ω ∂μ, ContinuousOn (fun t => H t ω) S) |
| {K : ℝ} (hK : 0 ≤ K) (hbound : ∀ t, ∀ᵐ ω ∂μ, |H t ω| ≤ K) |
| (s t : ℕ → ℕ → I) (ht : ∀ n, Monotone (t n)) (N : ℕ → ℕ) |
| (hst : ∀ n i, s n i ≤ t n i) |
| (hs : ∀ n i, i < N n → s n i ∈ S) |
| (htmem : ∀ n i, i < N n → t n i ∈ S) |
| (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0)) |
| (hdist : ∀ n i, i < N n → dist (s n i) (t n i) ≤ δ n) |
| (hA : ∀ᵐ ω ∂μ, Monotone (fun u => A u ω)) |
| {C : ℝ} (htotal : ∀ n, ∀ᵐ ω ∂μ, A (t n (N n)) ω - A (t n 0) ω ≤ C) : |
| Tendsto (fun n => ∫ ω, (∑ i ∈ Finset.range (N n), |
| (H (s n i) ω - H (t n i) ω) * |
| (M (t n (i + 1)) ω - M (t n i) ω)) ^ 2 ∂μ) atTop (𝓝 0) := by |
| let E : ℕ → ℕ → Ω → ℝ := fun n i ω => H (s n i) ω - H (t n i) ω |
| let R : ℕ → Ω → ℝ := fun n ω => ContinuousCoefficientModulus.pairModulus |
| (fun i => H (s n i) ω) (fun i => H (t n i) ω) (N n) |
| have hm (u : I) : StronglyMeasurable (H u) := (hH u).mono (F.le u) |
| have hE (n i : ℕ) : StronglyMeasurable[F (t n i)] (E n i) := |
| ((hH (s n i)).mono (F.mono (hst n i))).sub (hH (t n i)) |
| have hbE (n i : ℕ) : ∀ᵐ ω ∂μ, |E n i ω| ≤ 2 * K := by |
| filter_upwards [hbound (s n i), hbound (t n i)] with ω hsω htω |
| exact (abs_sub _ _).trans (by linarith) |
| have hR2 (n : ℕ) : MemLp (R n) 2 μ := |
| ContinuousCoefficientModulus.memLp_pairModulus |
| (fun i => H (s n i)) (fun i => H (t n i)) |
| (fun i => hm (s n i)) (fun i => hm (t n i)) (N n) |
| (show 0 ≤ 2 * K by positivity) (fun i _ => hbE n i) 2 |
| have henv (n : ℕ) : ∀ᵐ ω ∂μ, ∀ i < N n, E n i ω ^ 2 ≤ R n ω ^ 2 := by |
| apply Filter.Eventually.of_forall |
| intro ω i hi |
| have hb := ContinuousCoefficientModulus.abs_sub_le_pairModulus |
| (fun j => H (s n j) ω) (fun j => H (t n j) ω) hi |
| simpa only [sq_abs] using |
| (sq_le_sq₀ (abs_nonneg _) (ContinuousCoefficientModulus.pairModulus_nonneg _ _ _)).2 hb |
| have hlim : Tendsto (fun n => ∫ ω, R n ω ^ 2 ∂μ) atTop (𝓝 0) := |
| ContinuousCoefficientModulus.tendsto_integral_pairModulus_pow hS hm hcH hK |
| (fun u _ => hbound u) s t N hs htmem δ hδ hdist 2 (by norm_num) |
| exact MartingaleBracketIsometry.tendsto_integral_sq_grid_transform hM hQ hM2 t ht E hE |
| (show 0 ≤ 2 * K by positivity) hbE N (fun n => (hR2 n).integrable_sq) |
| (fun _ => Filter.Eventually.of_forall (fun ω => sq_nonneg _)) henv hA htotal hlim |
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| end Problem367.MartingaleCoefficientApproximation |
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| end |
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| end OAI |
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