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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 | import OAI.Probability.FreeIsing.Ising.CoordinateIntegrals
import OAI.Probability.FreeIsing.Ising.CovarianceTimeL2
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.IsingInnovation
variable {d : ℕ} {A : Type*} [MeasurableSpace A]
theorem exists_covariance_coordinate_integrals
{beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ)
(ν : Measure A) {B : ℝ≥0 → A → ℝ} (hB : IsPreBrownianReal B ν)
(hmB : ∀ t, Measurable (B t)) (hcB : ∀ a, Continuous (fun t => B t a))
(v : TreeVertex d) (T : ℝ≥0) :
let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
let Q := (volume.restrict (Set.Ioc (0 : ℝ) (T : ℝ))).prod P
let F := ArrayPosteriorStationarity.filtration B hmB
∃ (X : ℕ → TreeVertex d → State d A → ℝ) (I : TreeVertex d → State d A → ℝ),
(∀ n u, MemLp (X n u) 2 P) ∧
(∀ n u, Tendsto (fun k => ∫ q,
(BrownianGrid.uniformSum (innovation beta B u) (finiteCovariance beta B v u n)
T k q - X n u q) ^ 2 ∂P) atTop (𝓝 0)) ∧
(∀ u, Measurable (I u)) ∧
(∀ u, AEStronglyMeasurable[F T] (I u) P) ∧
(∀ u, MemLp (I u) 2 P) ∧
(∀ u, (∫ q, I u q ∂P) = 0) ∧
(∀ u, Tendsto (fun n => ∫ q, (X n u q - I u q) ^ 2 ∂P) atTop (𝓝 0)) ∧
(∀ u, (∫ q, I u q ^ 2 ∂P) =
∫ p : ℝ × State d A, covariance beta B v u p.1.toNNReal p.2 ^ 2 ∂Q) ∧
(∀ u, (∫ q, posterior beta B v T q * I u q ∂P) =
∫ p : ℝ × State d A, covariance beta B v u p.1.toNNReal p.2 ^ 2 ∂Q) ∧
Pairwise (fun u w => ∫ q, I u q * I w q ∂P = 0) := by
classical
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
let : IsProbabilityMeasure ν := hB.isGaussianProcess.isProbabilityMeasure
let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
let τ := volume.restrict (Set.Ioc (0 : ℝ) (T : ℝ))
let Q := τ.prod P
let F := ArrayPosteriorStationarity.filtration (d := d) B hmB
let G (n : ℕ) (u : TreeVertex d) (p : ℝ × State d A) :=
finiteCovariance beta B v u n p.1.toNNReal p.2
let C (u : TreeVertex d) (p : ℝ × State d A) :=
covariance beta B v u p.1.toNNReal p.2
have hG (n : ℕ) (u : TreeVertex d) : MemLp (G n u) 2 Q :=
memLp_finiteCovariance_time beta B hmB hcB P τ v u n
have hC (u : TreeVertex d) : MemLp (C u) 2 Q :=
memLp_covariance_time beta B hmB hcB P τ v u
have hW (u : TreeVertex d) :=
(isBrownianReal_innovation hμ ν hB hmB hcB u).toIsPreBrownianReal
have hAd (u : TreeVertex d) :=
(innovation_martingale hμ ν hB hmB hcB u).stronglyAdapted
have hind (u : TreeVertex d) (s t : ℝ≥0) (hst : s ≤ t) :=
BrownianArrayFiltration.coordinate_increment_indep_past
(innovation_increment_indep_filtration hμ ν hB hmB hcB s t hst) u
have hH (n : ℕ) (u : TreeVertex d) :=
finiteCovariance_stronglyAdapted beta B hmB v u n
have hcH (n : ℕ) (u : TreeVertex d) :
∀ᵐ q ∂P, Continuous (fun t => finiteCovariance beta B v u n t q) :=
Eventually.of_forall (finiteCovariance_continuous beta B hcB v u n)
have hbH (n : ℕ) (u : TreeVertex d) :
∀ᵐ q ∂P, ∀ t, |finiteCovariance beta B v u n t q| ≤ 2 :=
Eventually.of_forall (fun q t => finiteCovariance_abs_le_two beta B v u n t q)
obtain ⟨X, htX, hX, hzX, haX⟩ :=
exists_finiteCovariance_integrals hμ ν hB hmB hcB v T
have hiso (n : ℕ) (u : TreeVertex d) :
(∫ q, X n u q ^ 2 ∂P) = ∫ p, G n u p ^ 2 ∂Q := by
calc
_ = ∫ r in (0 : ℝ)..(T : ℝ), ∫ q,
finiteCovariance beta B v u n r.toNNReal q ^ 2 ∂P :=
BrownianFiltrationGrid.uniform_meanSquare_limit_isometry
(hW u) (hAd u) (hind u) _ (hH n u) (hcH n u)
(by norm_num) (hbH n u) T (hX n u) (haX n u)
_ = _ := interval_integral_eq_timeSpace T (hG n u).integrable_sq
have hpair (u : TreeVertex d) (n k : ℕ) :
(∫ q, (X n u q - X k u q) ^ 2 ∂P) =
∫ p, (G n u p - G k u p) ^ 2 ∂Q := by
calc
_ = ∫ r in (0 : ℝ)..(T : ℝ), ∫ q,
(finiteCovariance beta B v u n r.toNNReal q -
finiteCovariance beta B v u k r.toNNReal q) ^ 2 ∂P :=
BrownianFiltrationGrid.uniform_meanSquare_difference_isometry
(hW u) (hAd u) (hind u) _ _ (hH n u) (hH k u) (hcH n u) (hcH k u)
(by norm_num) (by norm_num) (hbH n u) (hbH k u) T
(hX n u) (hX k u) (haX n u) (haX k u)
_ = _ := interval_integral_eq_timeSpace T ((hG n u).sub (hG k u)).integrable_sq
have htest (u : TreeVertex d) (n : ℕ) :
(∫ q, posterior beta B v T q * X n u q ∂P) = ∫ p, C u p * G n u p ∂Q := by
rw [finiteCovariance_integral_cross hμ ν hB hmB hcB v u n T (hX n u) (haX n u)]
have he := interval_integral_eq_timeSpace T ((hG n u).integrable_mul (hC u))
simpa only [G, C, Pi.mul_apply, mul_comm] using he
have hex (u : TreeVertex d) :=
FilteringRepresentationClosure.exists_terminal_limit_of_coefficient_approximation
F T (posterior_memLp beta B hmB hcB P v T)
(fun n => X n u) (fun n => G n u) (C u)
(fun n => hX n u) (fun n => (htX n u).aestronglyMeasurable)
(fun n => hzX n u) (fun n => hG n u) (hC u)
(finiteCovariance_tendsto_timeMeanSquare hμ ν hB hmB hcB v u T)
(hpair u) (fun n => hiso n u) (htest u)
choose I hmI htI hI hzI haI heI hcI using hex
refine ⟨X, I, hX, haX, hmI, htI, hI, hzI, haI, heI, hcI, ?_⟩
apply FilteringRepresentationClosure.pairwise_orthogonal_of_meanSquare_limits hX hI haI
intro n
exact finiteCovariance_integrals_orthogonal hμ ν hB hmB hcB v n T (hX n) (haX n)
end Problem367.IsingInnovation
end
end OAI
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