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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 | import OAI.Probability.FreeIsing.Brownian.FiltrationIntegral
import OAI.Probability.FreeIsing.Brownian.SpatialIntegral
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal BigOperators
namespace Problem367.SimpleIntegral
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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