| import OAI.Probability.FreeIsing.Brownian.FiltrationIntegral |
| import OAI.Probability.FreeIsing.Brownian.SpatialIntegral |
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| namespace OAI |
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|
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal BigOperators |
|
|
| namespace Problem367.SimpleIntegral |
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|
| theorem predictable_array_increments_orthogonal |
| {Ω ι : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P] |
| (F : Filtration ℕ mΩ) (H D : ι → ℕ → Ω → ℝ) |
| (hH : ∀ a k, StronglyMeasurable[F k] (H a k)) |
| (hD : ∀ a k, StronglyMeasurable[F (k + 1)] (D a k)) |
| (hDi : ∀ a k, Integrable (D a k) P) |
| (hHD : ∀ a k, MemLp (fun ω => H a k ω * D a k ω) 2 P) |
| (hz : ∀ a k, P[D a k | F k] =ᵐ[P] 0) |
| (hsame : ∀ k, Pairwise fun a b => |
| ∫ ω, (H a k ω * D a k ω) * (H b k ω * D b k ω) ∂P = 0) : |
| Pairwise fun p q : ι × ℕ => ∫ ω, |
| (H p.1 p.2 ω * D p.1 p.2 ω) * (H q.1 q.2 ω * D q.1 q.2 ω) ∂P = 0 := by |
| have hoff (a b : ι) (i j : ℕ) (hij : i < j) : |
| (∫ ω, (H a i ω * D a i ω) * (H b j ω * D b j ω) ∂P) = 0 := by |
| have hc : StronglyMeasurable[F j] (fun ω => (H a i ω * D a i ω) * H b j ω) := by |
| simpa only [Pi.mul_def] using |
| (((hH a i).mono (F.mono hij.le)).mul |
| ((hD a i).mono (F.mono (Nat.succ_le_of_lt hij)))).mul (hH b j) |
| have hp : Integrable ((fun ω => (H a i ω * D a i ω) * H b j ω) * D b j) P := by |
| simpa only [Pi.mul_def, mul_assoc] using (hHD a i).integrable_mul (hHD b j) |
| simpa only [mul_assoc] using |
| integral_mul_eq_zero_of_condExp_eq_zero (F.le j) hc (hDi b j) hp (hz b j) |
| intro p q hpq |
| rcases lt_trichotomy p.2 q.2 with h | h | h |
| · exact hoff p.1 q.1 p.2 q.2 h |
| · have hab : p.1 ≠ q.1 := fun ha => hpq (Prod.ext ha h) |
| simpa only [h] using hsame q.2 hab |
| · simpa only [mul_comm] using hoff q.1 p.1 q.2 p.2 h |
|
|
| theorem integral_mul_sums_eq_zero |
| {Ω ι κ : Type*} [MeasurableSpace Ω] {P : Measure Ω} |
| (X : ι → Ω → ℝ) (Y : κ → Ω → ℝ) |
| (hX : ∀ i, MemLp (X i) 2 P) (hY : ∀ j, MemLp (Y j) 2 P) |
| (horth : ∀ i j, ∫ ω, X i ω * Y j ω ∂P = 0) |
| (s : Finset ι) (t : Finset κ) : |
| (∫ ω, (∑ i ∈ s, X i ω) * (∑ j ∈ t, Y j ω) ∂P) = 0 := by |
| have hint (i : ι) (j : κ) : Integrable (fun ω => X i ω * Y j ω) P := |
| (hX i).integrable_mul (hY j) |
| simp_rw [Finset.sum_mul_sum] |
| rw [integral_finsetSum s (fun i _ => |
| integrable_finsetSum t (fun j _ => hint i j))] |
| apply Finset.sum_eq_zero |
| intro i hi |
| rw [integral_finsetSum t (fun j _ => hint i j)] |
| exact Finset.sum_eq_zero fun j _ => horth i j |
|
|
| end Problem367.SimpleIntegral |
|
|
| namespace Problem367.BrownianArrayIntegral |
|
|
| theorem planted_grid_orthogonal |
| {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)) |
| (t : ℕ → ℝ≥0) (ht : Monotone t) |
| (H : ι → ℕ → S × (ι → Ω) → ℝ) |
| (hH : ∀ a k, StronglyMeasurable[signalArrayFiltration B hmB (t k)] (H a k)) |
| (hH2 : ∀ a k, MemLp (H a k) 2 (ν.prod (Measure.infinitePi μ))) : |
| Pairwise fun p q : ι × ℕ => ∫ x : S × (ι → Ω), |
| (H p.1 p.2 x * (B p.1 (t (p.2 + 1)) (x.2 p.1) - B p.1 (t p.2) (x.2 p.1))) * |
| (H q.1 q.2 x * (B q.1 (t (q.2 + 1)) (x.2 q.1) - B q.1 (t q.2) (x.2 q.1))) |
| ∂ν.prod (Measure.infinitePi μ) = 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 G : Filtration ℕ (inferInstance : MeasurableSpace (S × (ι → Ω))) := |
| ⟨fun k => F (t k), fun _ _ h => F.mono (ht h), fun k => F.le (t k)⟩ |
| let D : ι → ℕ → S × (ι → Ω) → ℝ := |
| fun a k x => B a (t (k + 1)) (x.2 a) - B a (t k) (x.2 a) |
| have hC (a : ι) := planted_coordinate_preBrownian ν μ hB a |
| have hAd (a : ι) := (preBrownian_martingale_signalArrayFiltration ν μ hB hmB a).stronglyAdapted |
| have hD2 (a : ι) (k : ℕ) : MemLp (D a k) 2 P := |
| ((hC a).isGaussianProcess.hasGaussianLaw_eval (t (k + 1))).memLp_two.sub |
| ((hC a).isGaussianProcess.hasGaussianLaw_eval (t k)).memLp_two |
| have hind (k : ℕ) : IndepFun (fun x a => H a k x) (fun x a => D a k x) P := |
| PredictableL2.planted_coefficients_indep_increments ν μ hB hmB |
| (ht (Nat.le_succ k)) (fun a => hH a k) |
| have hHD (a : ι) (k : ℕ) : MemLp (fun x => H a k x * D a k x) 2 P := |
| PredictableL2.memLp_two_mul_of_indep |
| ((hind k).comp (measurable_pi_apply a) (measurable_pi_apply a)) (hH2 a k) (hD2 a k) |
| have hz (a : ι) (k : ℕ) : (∫ x, D a k x ∂P) = 0 := by |
| change (∫ x : S × (ι → Ω), B a (t (k + 1)) (x.2 a) - B a (t k) (x.2 a) ∂P) = 0 |
| rw [integral_sub ((hC a).integrable_eval _) ((hC a).integrable_eval _), |
| (hC a).integral_eval, (hC a).integral_eval, sub_self] |
| have hnoise (k : ℕ) : Pairwise fun a b => ∫ x, D a k x * D b k x ∂P = 0 := by |
| intro a b hab |
| have hi := (planted_coordinate_increments_indep ν μ hB hmB (t k) (t (k + 1)) hab).integral_fun_mul_eq_mul_integral |
| (hD2 a k).aestronglyMeasurable (hD2 b k).aestronglyMeasurable |
| change (∫ x, D a k x * D b k x ∂P) = (∫ x, D a k x ∂P) * ∫ x, D b k x ∂P at hi |
| simpa only [hz, zero_mul] using hi |
| apply SimpleIntegral.predictable_array_increments_orthogonal G H D hH |
| · intro a k |
| exact (hAd a (t (k + 1))).sub ((hAd a (t k)).mono (F.mono (ht (Nat.le_succ k)))) |
| · exact fun a k => (hD2 a k).integrable (by norm_num) |
| · exact hHD |
| · intro a k |
| apply BrownianFiltration.increment_condExp_zero (hC a) (hAd a) |
| change Indep _ (signalArrayFiltration B hmB (t k)) P |
| rw [signalArrayFiltration_eq_comap] |
| exact (planted_array_increments_indep_signal_past ν μ hB hmB (ht (Nat.le_succ k))).comp |
| (measurable_pi_apply a) measurable_id |
| · intro k |
| exact PredictableL2.weighted_array_orthogonal (hind k) (fun a => hH2 a k) |
| (fun a => hD2 a k) (hnoise k) |
|
|
| theorem planted_grid_sums_orthogonal |
| {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)) |
| (t : ℕ → ℝ≥0) (ht : Monotone t) |
| (H : ι → ℕ → S × (ι → Ω) → ℝ) |
| (hH : ∀ a k, StronglyMeasurable[signalArrayFiltration B hmB (t k)] (H a k)) |
| (hH2 : ∀ a k, MemLp (H a k) 2 (ν.prod (Measure.infinitePi μ))) |
| {a b : ι} (hab : a ≠ b) (u v : Finset ℕ) : |
| (∫ x : S × (ι → Ω), |
| (∑ k ∈ u, H a k x * (B a (t (k + 1)) (x.2 a) - B a (t k) (x.2 a))) * |
| (∑ k ∈ v, H b k x * (B b (t (k + 1)) (x.2 b) - B b (t k) (x.2 b))) |
| ∂ν.prod (Measure.infinitePi μ)) = 0 := by |
| let P := ν.prod (Measure.infinitePi μ) |
| let D : ι → ℕ → S × (ι → Ω) → ℝ := |
| fun a k x => B a (t (k + 1)) (x.2 a) - B a (t k) (x.2 a) |
| have hHD (a : ι) (k : ℕ) : MemLp (fun x => H a k x * D a k x) 2 P := by |
| have hc := planted_coordinate_preBrownian ν μ hB a |
| have hd : MemLp (D a k) 2 P := |
| (hc.isGaussianProcess.hasGaussianLaw_eval (t (k + 1))).memLp_two.sub |
| (hc.isGaussianProcess.hasGaussianLaw_eval (t k)).memLp_two |
| have hi := PredictableL2.planted_coefficients_indep_increments ν μ hB hmB |
| (ht (Nat.le_succ k)) (fun a => hH a k) |
| exact PredictableL2.memLp_two_mul_of_indep |
| (hi.comp (measurable_pi_apply a) (measurable_pi_apply a)) (hH2 a k) hd |
| apply SimpleIntegral.integral_mul_sums_eq_zero |
| (fun k x => H a k x * D a k x) (fun k x => H b k x * D b k x) |
| (hHD a) (hHD b) _ u v |
| intro k l |
| exact planted_grid_orthogonal ν μ hB hmB t ht H hH hH2 |
| (show (a, k) ≠ (b, l) from fun h => hab (congrArg Prod.fst h)) |
|
|
| end Problem367.BrownianArrayIntegral |
|
|
| end |
|
|
| end OAI |
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|
