| import OAI.Probability.FreeIsing.Ising.CoordinateIntegrals |
| import OAI.Probability.FreeIsing.Ising.CovarianceTimeL2 |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal Topology |
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| namespace Problem367.IsingInnovation |
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| variable {d : ℕ} {A : Type*} [MeasurableSpace A] |
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| 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) |
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| end Problem367.IsingInnovation |
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| end |
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| end OAI |
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