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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 | import OAI.Probability.FreeIsing.Brownian.ContinuousRowCompletion
import OAI.Probability.FreeIsing.Gibbs.CovarianceNativeRowCauchy
import OAI.Probability.FreeIsing.Ising.InnovationIndependence
import OAI.Probability.FreeIsing.Ising.MixedFiltering
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology BigOperators
namespace Problem367.IsingInnovation
variable {d : ℕ} {A : Type*} [MeasurableSpace A]
theorem finiteCovariance_fullTime_interval_energy_cauchy
(hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(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} (hT : 0 ≤ T) :
let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
(∀ n, Summable (fun u => ∫ t in (0 : ℝ)..T,
∫ q, finiteCovariance beta B v u n t.toNNReal q ^ 2 ∂P)) ∧
(∀ n k, Summable (fun u => ∫ t in (0 : ℝ)..T,
∫ q, (finiteCovariance beta B v u n t.toNNReal q -
finiteCovariance beta B v u k t.toNNReal q) ^ 2 ∂P)) ∧
∀ ε > (0 : ℝ), ∃ N : ℕ, ∀ n ≥ N, ∀ k ≥ N,
(∑' u, ∫ t in (0 : ℝ)..T,
∫ q, (finiteCovariance beta B v u n t.toNNReal q -
finiteCovariance beta B v u k t.toNNReal q) ^ 2 ∂P) < ε := by
have h := hμ.finiteGibbsCovariance_fullTime_interval_energy_cauchy
hd hb hcrit (fun _ : TreeVertex d => ν) (fun _ => hB)
(fun _ => hmB) (fun _ => hcB) v hT
simpa only [finiteCovariance, plantedArrayObservation,
ArrayPosteriorStationarity.spinArray, add_comm] using h
theorem exists_continuous_covariance_integral
(hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(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) :
let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
let F := ArrayPosteriorStationarity.filtration B hmB
∃ (I : ℕ → TreeVertex d → ℝ≥0 → State d A → ℝ)
(Y : ℕ → ℝ≥0 → State d A → ℝ) (Z : ℝ≥0 → State d A → ℝ)
(hI2 : ∀ n u t, MemLp (I n u t) 2 P)
(hY2 : ∀ n t, MemLp (Y n t) 2 P) (_hZ2 : ∀ t, MemLp (Z t) 2 P),
(∀ n, IsStronglyPredictable F (Y n) ∧ Martingale (Y n) F P ∧
(∀ᵐ q ∂P, Continuous (fun t => Y n t q))) ∧
(∀ n u (N : ℕ), 0 < N → Tendsto (fun k => ∫ q,
(BrownianGrid.uniformSum (innovation beta B u) (finiteCovariance beta B v u n)
N k q - I n u N q) ^ 2 ∂P) atTop (𝓝 0)) ∧
(∀ n t, HasSum (fun u => (hI2 n u t).toLp (I n u t))
((hY2 n t).toLp (Y n t))) ∧
(∀ n t, (∫ q, Y n t q ^ 2 ∂P) =
∑' u, ∫ s in (0 : ℝ)..(t : ℝ),
∫ q, finiteCovariance beta B v u n s.toNNReal q ^ 2 ∂P) ∧
(∀ n m (N : ℕ), 0 < N → (∫ q, (Y n N q - Y m N q) ^ 2 ∂P) =
∑' u, ∫ s in (0 : ℝ)..(N : ℝ), ∫ q,
(finiteCovariance beta B v u n s.toNNReal q -
finiteCovariance beta B v u m s.toNNReal q) ^ 2 ∂P) ∧
IsStronglyPredictable F Z ∧ Martingale Z F P ∧
(∀ᵐ q ∂P, Continuous (fun t => Z t q)) ∧ Z 0 =ᵐ[P] 0 ∧
(∀ t, (∫ q, Z t q ∂P) = 0) ∧
(∀ t, Tendsto (fun n => ∫ q, (Y n t q - Z t q) ^ 2 ∂P) atTop (𝓝 0)) ∧
(∀ t : ℝ≥0, Tendsto (fun n => ∑' u, ∫ s in (0 : ℝ)..(t : ℝ),
∫ q, finiteCovariance beta B v u n s.toNNReal q ^ 2 ∂P) atTop
(𝓝 (∫ q, Z t q ^ 2 ∂P))) ∧
(∀ T, Tendsto (fun n => ∫ q,
MartingaleStrongMaximal.maximalSquare (Y n - Z) T q ∂P) atTop (𝓝 0)) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
let : IsProbabilityMeasure ν := hB.isGaussianProcess.isProbabilityMeasure
let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
have hW (u : TreeVertex d) := isBrownianReal_innovation hμ ν hB hmB hcB u
have hAd (u : TreeVertex d) :=
(innovation_martingale hμ ν hB hmB hcB u).stronglyAdapted
have hcross (s t : ℝ≥0) (hst : s ≤ t) : Pairwise fun u w =>
IndepFun (fun q => innovation beta B u t q - innovation beta B u s q)
(fun q => innovation beta B w t q - innovation beta B w s q) P := by
intro u w huw
exact (innovation_increment_law_and_indep hμ ν hB hmB hcB
(Y := fun _ => ()) s t hst measurable_const).2.1.indepFun huw
have hrow (N : ℕ) := finiteCovariance_fullTime_interval_energy_cauchy
hd hμ hb hcrit ν hB hmB hcB v (Nat.cast_nonneg N)
exact BrownianArrayFiltration.exists_global_continuous_integral_limit_of_row_energy_cauchy
(fun u => (hW u).toIsPreBrownianReal) hAd
(innovation_increment_indep_filtration hμ ν hB hmB hcB) hcross
(fun u => (hW u).cont)
(fun n u => finiteCovariance beta B v u n)
(fun n u => finiteCovariance_stronglyAdapted beta B hmB v u n)
(fun n u => Eventually.of_forall (finiteCovariance_continuous beta B hcB v u n))
(fun _ _ => 2) (fun _ _ => by norm_num)
(fun n u => Eventually.of_forall (fun q t => finiteCovariance_abs_le_two beta B v u n t q))
(fun n N => (hrow N).1 n) (fun N _hN => (hrow N).2.2)
end Problem367.IsingInnovation
end
end OAI
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