| import OAI.Probability.FreeIsing.Brownian.ArraySimpleIntegral |
| import OAI.Probability.FreeIsing.Brownian.FiltrationIntegralProperties |
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| 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 |
| |
