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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 | import OAI.Probability.FreeIsing.Estimates.CoefficientTerminalCompletion
import OAI.Probability.FreeIsing.Ising.InnovationIndependence
import OAI.Probability.FreeIsing.Ising.MixedFiltering
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.IsingInnovation
variable {d : ℕ} {A : Type*} [MeasurableSpace A]
lemma posterior_memLp
(beta : ℝ) (B : ℝ≥0 → A → ℝ) (hmB : ∀ t, Measurable (B t))
(hcB : ∀ a, Continuous (fun t => B t a))
(P : Measure (State d A)) [IsFiniteMeasure P] (v : TreeVertex d) (T : ℝ≥0) :
MemLp (posterior beta B v T) 2 P := by
apply MemLp.of_bound
((measurable_posterior_joint beta B hmB hcB v).comp
(measurable_const.prodMk measurable_id)).aestronglyMeasurable 1
exact Eventually.of_forall fun q => by
simpa only [Real.norm_eq_abs, Function.comp_apply, Prod.fst, Prod.snd, id_eq] using
posterior_abs_le_one beta B v T q
lemma finiteCovariance_memLp
(beta : ℝ) (B : ℝ≥0 → A → ℝ) (hmB : ∀ t, Measurable (B t))
(P : Measure (State d A)) [IsFiniteMeasure P]
(v u : TreeVertex d) (n : ℕ) (t : ℝ≥0) :
MemLp (finiteCovariance beta B v u n t) 2 P := by
apply MemLp.of_bound
(((finiteCovariance_stronglyAdapted beta B hmB v u n) t).mono
(ArrayPosteriorStationarity.filtration B hmB |>.le t)).aestronglyMeasurable 2
exact Eventually.of_forall fun q => by
simpa only [Real.norm_eq_abs] using finiteCovariance_abs_le_two beta B v u n t q
theorem exists_finiteCovariance_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 F := ArrayPosteriorStationarity.filtration B hmB
∃ X : ℕ → TreeVertex d → State d A → ℝ,
(∀ n u, StronglyMeasurable[F T] (X n u)) ∧
(∀ n u, MemLp (X n u) 2 P) ∧
(∀ n u, (∫ q, X n u q ∂P) = 0) ∧
(∀ 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)) := by
classical
let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
have hex (n : ℕ) (u : TreeVertex d) :=
BrownianFiltrationGrid.exists_terminal_uniform_integral_of_bounded_continuous
(isBrownianReal_innovation hμ ν hB hmB hcB u).toIsPreBrownianReal
(innovation_martingale hμ ν hB hmB hcB u).stronglyAdapted
(fun s t hst => BrownianArrayFiltration.coordinate_increment_indep_past
(innovation_increment_indep_filtration hμ ν hB hmB hcB s t hst) u)
(finiteCovariance beta B v u n) (finiteCovariance_stronglyAdapted beta B hmB v u n)
(Eventually.of_forall (finiteCovariance_continuous beta B hcB v u n))
(show (0 : ℝ) ≤ 2 by norm_num)
(Eventually.of_forall (fun q t => finiteCovariance_abs_le_two beta B v u n t q)) T
choose X htX hX hzX haX using hex
exact ⟨X, htX, hX, hzX, haX⟩
lemma interval_integral_eq_timeSpace
{Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [SFinite P]
(T : ℝ≥0) {f : ℝ × Ω → ℝ}
(hf : Integrable f ((volume.restrict (Set.Ioc (0 : ℝ) (T : ℝ))).prod P)) :
(∫ r in (0 : ℝ)..(T : ℝ), ∫ q, f (r, q) ∂P) =
∫ p, f p ∂(volume.restrict (Set.Ioc (0 : ℝ) (T : ℝ))).prod P := by
rw [intervalIntegral.integral_of_le T.coe_nonneg]
exact (integral_prod f hf).symm
theorem finiteCovariance_integral_cross
{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 u : TreeVertex d) (n : ℕ) (T : ℝ≥0) {X : State d A → ℝ}
(hX : MemLp X 2 (μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))))
(haX : Tendsto (fun k => ∫ q,
(BrownianGrid.uniformSum (innovation beta B u) (finiteCovariance beta B v u n)
T k q - X q) ^ 2 ∂μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)))
atTop (𝓝 0)) :
(∫ q, posterior beta B v T q * X q
∂μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))) =
∫ r in (0 : ℝ)..(T : ℝ), ∫ q,
finiteCovariance beta B v u n r.toNNReal q * covariance beta B v u r.toNNReal q
∂μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
let : IsProbabilityMeasure ν := hB.isGaussianProcess.isProbabilityMeasure
apply FilteringRepresentationClosure.integral_mul_eq_of_meanSquare_limit
(posterior_memLp beta B hmB hcB _ v T) hX _ haX
(tendsto_finiteCovariance_cross_uniformSum hμ ν hB hmB hcB v u n T)
exact BrownianFiltrationGrid.uniformSum_memLp
(isBrownianReal_innovation hμ ν hB hmB hcB u).toIsPreBrownianReal
(innovation_martingale hμ ν hB hmB hcB u).stronglyAdapted
(fun s t hst => BrownianArrayFiltration.coordinate_increment_indep_past
(innovation_increment_indep_filtration hμ ν hB hmB hcB s t hst) u)
(finiteCovariance beta B v u n) (finiteCovariance_stronglyAdapted beta B hmB v u n)
(finiteCovariance_memLp beta B hmB _ v u n) T
theorem finiteCovariance_integrals_orthogonal
{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) (n : ℕ) (T : ℝ≥0) {X : TreeVertex d → State d A → ℝ}
(hX : ∀ u, MemLp (X u) 2 (μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))))
(haX : ∀ u, Tendsto (fun k => ∫ q,
(BrownianGrid.uniformSum (innovation beta B u) (finiteCovariance beta B v u n)
T k q - X u q) ^ 2 ∂μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)))
atTop (𝓝 0)) :
Pairwise fun u w => ∫ q, X u q * X w q
∂μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν)) = 0 := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
let : IsProbabilityMeasure ν := hB.isGaussianProcess.isProbabilityMeasure
apply BrownianArrayFiltration.uniform_meanSquare_limits_orthogonal
(fun u => (isBrownianReal_innovation hμ ν hB hmB hcB u).toIsPreBrownianReal)
(fun u => (innovation_martingale hμ ν hB hmB hcB u).stronglyAdapted)
(innovation_increment_indep_filtration hμ ν hB hmB hcB) _
(fun u => finiteCovariance beta B v u n)
(fun u => finiteCovariance_stronglyAdapted beta B hmB v u n)
(fun u => finiteCovariance_memLp beta B hmB _ v u n) T hX haX
intro s t hst 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
end Problem367.IsingInnovation
end
end OAI
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