File size: 3,292 Bytes
b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | import OAI.Probability.FreeIsing.Estimates.ContinuousCoefficientModulus
import OAI.Probability.FreeIsing.Martingales.BracketIsometry
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology
namespace Problem367.MartingaleCoefficientApproximation
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω}
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
end Problem367.MartingaleCoefficientApproximation
end
end OAI
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