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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 | import OAI.Probability.FreeIsing.Brownian.IntegralProcess
import OAI.Probability.FreeIsing.Brownian.IntegralTransport
import OAI.Probability.FreeIsing.Ising.CanonicalPosteriorStops
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation IsingCopyEnergy BrownianIntegralProcess
variable {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
[IsProbabilityMeasure μ]
theorem ae_innovationMap_projection (j : Bool) :
∀ᵐ q ∂copies beta μ, innovationMap beta (projection j q) = q.1 := by
have h := IsingConditionalCopyBounds.ae_innovationMap_copies
(beta := beta) (μ := μ)
(rho := (observationLaw μ).map (innovationMap beta))
(eta := kernel beta μ) kernel_graphLaw
filter_upwards [h] with q hq
cases j
· exact hq.2
· exact hq.1
theorem stoppedUniformSum_copy_eq_ae (j : Bool) (v : TreeVertex d)
(H : ℝ≥0 → Paths d → ℝ) :
∀ᵐ q ∂copies beta μ, ∀ (T : ℝ≥0) (n : ℕ) (t : ℝ≥0),
stoppedUniformSum (fun s (p : Joint d) => p.1 v s)
(fun s p => H s (projection j p)) T n t q =
stoppedUniformSum (fun s x => innovationMap beta x v s) H T n t
(projection j q) := by
filter_upwards [ae_innovationMap_projection (beta := beta) (μ := μ) j] with q hq
intro T n t
simp only [stoppedUniformSum, StoppedSimpleIntegral.intervalSum, hq]
theorem tendsto_stoppedUniformSum_copy (j : Bool) (v : TreeVertex d)
{H I : ℝ≥0 → Paths d → ℝ}
(hH : ∀ s, StronglyMeasurable (H s)) (T t : ℝ≥0)
(hI : AEStronglyMeasurable (I t) (observationLaw μ))
(hlim : Tendsto (fun n => ∫ x,
(stoppedUniformSum (fun s x => innovationMap beta x v s) H T n t x - I t x) ^ 2
∂observationLaw μ) atTop (𝓝 0)) :
Tendsto (fun n => ∫ q,
(stoppedUniformSum (fun s (p : Joint d) => p.1 v s)
(fun s p => H s (projection j p)) T n t q - I t (projection j q)) ^ 2
∂copies beta μ) atTop (𝓝 0) := by
have hB (s : ℝ≥0) : Measurable (fun x : Paths d => innovationMap beta x v s) :=
(continuous_eval_const s).measurable.comp
((measurable_pi_apply v).comp (measurable_innovationMap beta))
have hm (n : ℕ) : AEStronglyMeasurable
(stoppedUniformSum (fun s x => innovationMap beta x v s) H T n t)
(observationLaw μ) := by
apply Measurable.aestronglyMeasurable
unfold stoppedUniformSum StoppedSimpleIntegral.intervalSum
exact Finset.measurable_sum _ (fun i _ =>
(hH _).measurable.mul ((hB _).sub (hB _)))
have hp := BrownianGrid.meanSquare_limit_comp
(copyProjection_measurePreserving beta μ j) hm hI hlim
have heq (n : ℕ) :
(∫ q, (stoppedUniformSum (fun s (p : Joint d) => p.1 v s)
(fun s p => H s (projection j p)) T n t q - I t (projection j q)) ^ 2
∂copies beta μ) =
∫ q, (stoppedUniformSum (fun s x => innovationMap beta x v s) H T n t
(projection j q) - I t (projection j q)) ^ 2 ∂copies beta μ := by
apply integral_congr_ae
filter_upwards [stoppedUniformSum_copy_eq_ae (beta := beta) (μ := μ) j v H] with q hq
rw [hq T n t]
simpa only [heq] using hp
theorem integer_stoppedUniformSum_limits_copy (j : Bool) (v : TreeVertex d)
{H I : ℝ≥0 → Paths d → ℝ}
(hH : StronglyAdapted (filtration d) H)
(hI : ∀ t, MemLp (I t) 2 (observationLaw μ))
(hlim : ∀ N : ℕ, 0 < N → ∀ t ≤ (N : ℝ≥0),
Tendsto (fun n => ∫ x,
(stoppedUniformSum (fun s x => innovationMap beta x v s) H N n t x - I t x) ^ 2
∂observationLaw μ) atTop (𝓝 0)) :
∀ N : ℕ, 0 < N → ∀ t ≤ (N : ℝ≥0),
Tendsto (fun n => ∫ q,
(stoppedUniformSum (fun s (p : Joint d) => p.1 v s)
(fun s p => H s (projection j p)) N n t q - I t (projection j q)) ^ 2
∂copies beta μ) atTop (𝓝 0) := by
intro N hN t ht
exact tendsto_stoppedUniformSum_copy j v (fun s => (hH s).mono ((filtration d).le s))
N t (hI t).aestronglyMeasurable (hlim N hN t ht)
end Problem367.IsingObservationFactor
end
end OAI
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