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f61e547 | 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 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | import OAI.Probability.FreeIsing.Brownian.ArraySimpleIntegral
import OAI.Probability.FreeIsing.Brownian.FiltrationIntegralProperties
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology BigOperators
namespace Problem367.BrownianArrayIntegral
open BrownianGrid BrownianFiltrationGrid
theorem exists_planted_channel_integrals
{ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω]
(ν : Measure S) (μ : ι → Measure Ω) [IsProbabilityMeasure ν]
{B : ι → ℝ≥0 → Ω → ℝ} (hB : ∀ a, IsPreBrownianReal (B a) (μ a))
(hmB : ∀ a t, Measurable (B a t))
(H : ι → ℝ≥0 → S × (ι → Ω) → ℝ)
(hH : ∀ a, StronglyAdapted (signalArrayFiltration B hmB) (H a))
(hc : ∀ a, ∀ᵐ x ∂ν.prod (Measure.infinitePi μ), Continuous (fun t => H a t x))
(C : ι → ℝ) (hC : ∀ a, 0 ≤ C a)
(hb : ∀ a, ∀ᵐ x ∂ν.prod (Measure.infinitePi μ), ∀ t, |H a t x| ≤ C a)
(T : ℝ≥0) :
∃ I : ι → S × (ι → Ω) → ℝ,
(∀ a, MemLp (I a) 2 (ν.prod (Measure.infinitePi μ))) ∧
(∀ a, StronglyMeasurable (I a)) ∧
(∀ a, AEStronglyMeasurable[signalArrayFiltration B hmB T]
(I a) (ν.prod (Measure.infinitePi μ))) ∧
(∀ a, (∫ x, I a x ∂ν.prod (Measure.infinitePi μ)) = 0) ∧
(∀ a, (∫ x, I a x ^ 2 ∂ν.prod (Measure.infinitePi μ)) =
∫ t in (0 : ℝ)..(T : ℝ), ∫ x, H a t.toNNReal x ^ 2 ∂ν.prod (Measure.infinitePi μ)) ∧
(Pairwise fun a b => ∫ x, I a x * I b x ∂ν.prod (Measure.infinitePi μ) = 0) ∧
(∀ a, Tendsto (fun n => ∫ x,
(BrownianGrid.uniformSum (fun t x => B a t (x.2 a)) (H a) T n x - I a x) ^ 2
∂ν.prod (Measure.infinitePi μ)) atTop (𝓝 0)) := by
have (a : ι) : IsProbabilityMeasure (μ a) := (hB a).isGaussianProcess.isProbabilityMeasure
let P := ν.prod (Measure.infinitePi μ)
let F := signalArrayFiltration (S := S) B hmB
let W : ι → ℝ≥0 → S × (ι → Ω) → ℝ := fun a t x => B a t (x.2 a)
have hW (a : ι) : IsPreBrownianReal (W a) P := planted_coordinate_preBrownian ν μ hB a
have hAd (a : ι) : StronglyAdapted F (W a) :=
(preBrownian_martingale_signalArrayFiltration ν μ hB hmB a).stronglyAdapted
have hi (a : ι) (s t : ℝ≥0) (hst : s ≤ t) :
Indep (MeasurableSpace.comap (fun x => W a t x - W a s x) inferInstance) (F s) P := by
change Indep _ (signalArrayFiltration B hmB s) P
rw [signalArrayFiltration_eq_comap]
exact (planted_array_increments_indep_signal_past ν μ hB hmB hst).comp
(measurable_pi_apply a) measurable_id
have hmH (a : ι) (t : ℝ≥0) : AEStronglyMeasurable (H a t) P :=
((hH a t).mono (F.le t)).aestronglyMeasurable
have hH2 (a : ι) (t : ℝ≥0) : MemLp (H a t) 2 P := by
apply MemLp.of_bound (hmH a t) (C a)
filter_upwards [hb a] with x hx
simpa only [Real.norm_eq_abs] using hx t
have hex (a : ι) := BrownianFiltrationGrid.exists_uniformSumLp_limit
(hW a) (hAd a) (hi a) (H a) (hH a) (hH2 a) T
(fun _ hε => MeanSquareContinuity.uniform_integral_sq_sub_le (hmH a) (hc a) (hC a) (hb a) T hε)
choose I hI using hex
have hprop (a : ι) := BrownianFiltrationGrid.uniformSumLp_limit_terminal_centered
(hW a) (hAd a) (hi a) (H a) (hH a) (hH2 a) T (hI a)
refine ⟨fun a => I a, fun a => Lp.memLp (I a), fun a => Lp.stronglyMeasurable (I a),
fun a => (hprop a).1, fun a => (hprop a).2, ?_, ?_, ?_⟩
· intro a
exact BrownianFiltrationGrid.uniformSumLp_limit_isometry
(hW a) (hAd a) (hi a) (H a) (hH a) (hH2 a) (hc a) (hC a) (hb a) T (hI a)
· intro a b hab
have hlim := (hI a).inner (𝕜 := ℝ) (hI b)
have hz (n : ℕ) : inner ℝ
(BrownianFiltrationGrid.uniformSumLp (hW a) (hAd a) (hi a) (H a) (hH a) (hH2 a) T n)
(BrownianFiltrationGrid.uniformSumLp (hW b) (hAd b) (hi b) (H b) (hH b) (hH2 b) T n) = 0 := by
rw [BrownianFiltrationGrid.uniformSumLp, BrownianFiltrationGrid.uniformSumLp,
PredictableL2.inner_toLp]
exact planted_grid_sums_orthogonal ν μ hB hmB (uniformGrid T n) (uniformGrid_mono T n)
(fun a k => H a (uniformGrid T n k)) (fun a k => hH a _) (fun a k => hH2 a _)
hab (Finset.range n) (Finset.range n)
have heq : inner ℝ (I a) (I b) = 0 :=
tendsto_nhds_unique hlim (by simpa only [hz] using
(tendsto_const_nhds : Tendsto (fun _ : ℕ => (0 : ℝ)) atTop (𝓝 0)))
rw [L2.inner_def] at heq
simpa only [Real.inner_apply] using heq
· intro a
exact BrownianFiltrationGrid.integral_sq_error_tendsto_of_Lp_tendsto
(hW a) (hAd a) (hi a) (H a) (hH a) (hH2 a) T (hI a)
theorem exists_planted_infinite_integral
{ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω]
(ν : Measure S) (μ : ι → Measure Ω) [IsProbabilityMeasure ν]
{B : ι → ℝ≥0 → Ω → ℝ} (hB : ∀ a, IsPreBrownianReal (B a) (μ a))
(hmB : ∀ a t, Measurable (B a t))
(H : ι → ℝ≥0 → S × (ι → Ω) → ℝ)
(hH : ∀ a, StronglyAdapted (signalArrayFiltration B hmB) (H a))
(hc : ∀ a, ∀ᵐ x ∂ν.prod (Measure.infinitePi μ), Continuous (fun t => H a t x))
(C : ι → ℝ) (hC : ∀ a, 0 ≤ C a)
(hb : ∀ a, ∀ᵐ x ∂ν.prod (Measure.infinitePi μ), ∀ t, |H a t x| ≤ C a)
(T : ℝ≥0)
(henergy : Summable fun a =>
∫ t in (0 : ℝ)..(T : ℝ), ∫ x, H a t.toNNReal x ^ 2 ∂ν.prod (Measure.infinitePi μ)) :
∃ (I : ι → S × (ι → Ω) → ℝ) (Y : S × (ι → Ω) → ℝ),
(∀ a, MemLp (I a) 2 (ν.prod (Measure.infinitePi μ))) ∧
(∀ a, Tendsto (fun n => ∫ x,
(BrownianGrid.uniformSum (fun t x => B a t (x.2 a)) (H a) T n x - I a x) ^ 2
∂ν.prod (Measure.infinitePi μ)) atTop (𝓝 0)) ∧
Measurable Y ∧ MemLp Y 2 (ν.prod (Measure.infinitePi μ)) ∧
(∫ x, Y x ∂ν.prod (Measure.infinitePi μ)) = 0 ∧
(∫ x, Y x ^ 2 ∂ν.prod (Measure.infinitePi μ)) =
∑' a, ∫ t in (0 : ℝ)..(T : ℝ), ∫ x, H a t.toNNReal x ^ 2 ∂ν.prod (Measure.infinitePi μ) ∧
Tendsto (fun s : Finset ι => ∫ x,
((∑ a ∈ s, I a x) - Y x) ^ 2 ∂ν.prod (Measure.infinitePi μ)) atTop (𝓝 0) := by
have (a : ι) : IsProbabilityMeasure (μ a) := (hB a).isGaussianProcess.isProbabilityMeasure
obtain ⟨I, hI2, _, _, hz, hiso, horth, happrox⟩ :=
exists_planted_channel_integrals ν μ hB hmB H hH hc C hC hb T
obtain ⟨Y, hY, hmY, hsY, heY⟩ := PredictableL2.exists_orthogonal_sum I hI2 horth
(by simpa only [hiso] using henergy)
refine ⟨I, Y, hI2, happrox, hmY, hY, ?_, ?_, ?_⟩
· exact PredictableL2.integral_eq_zero_of_hasSum I hI2 hY hsY hz
· simpa only [hiso] using heY
· exact PredictableL2.tendsto_integral_sq_error_of_hasSum I hI2 hY hsY
end Problem367.BrownianArrayIntegral
end
end OAI
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