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a89d282 | 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 | import OAI.Probability.InvariantIsing.Gaussian.GaussianGramTransformLimit
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
namespace OAI
/-! Convergence in probability of the physical Gaussian Gram resolvents. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped Topology ENNReal
namespace InvariantIsing
theorem gaussianPatternStieltjes_tendsto {α t : ℝ} (hα : 0 ≤ α) (ht : 0 < t)
{Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)
(Z : (N : ℕ) → Ω → EuclideanSpace ℝ (Fin N × Fin (gaussianPatternCount α N)))
(hZ : ∀ N, HasLaw (Z N) (stdGaussian _) P) :
TendstoInMeasure P (fun k ω => gaussianGramStieltjes t (Z (k+1) ω)) atTop
(fun _ => marchenkoPasturTransform t α) := by
have hi (k : ℕ) : Integrable (fun ω =>
gaussianGramStieltjes t (Z (k+1) ω)-marchenkoPasturTransform t α) P := by
have hm : AEStronglyMeasurable
(fun z : EuclideanSpace ℝ (Fin (k+1) × Fin (gaussianPatternCount α (k+1))) =>
gaussianGramStieltjes t z-marchenkoPasturTransform t α) (stdGaussian _) :=
((continuous_gaussianGramStieltjes (N := k+1)
(m := gaussianPatternCount α (k+1)) ht).sub continuous_const).aestronglyMeasurable
have hh : Integrable (fun z : EuclideanSpace ℝ (Fin (k+1) × Fin (gaussianPatternCount α (k+1))) =>
gaussianGramStieltjes t z-marchenkoPasturTransform t α) (stdGaussian _) :=
(integrable_norm_iff hm).mp (by
simpa only [Real.norm_eq_abs] using gaussianGramTransform_error_integrable ht α)
exact (hZ (k+1)).integrable_comp hh
have he (k : ℕ) : eLpNorm ((fun ω => gaussianGramStieltjes t (Z (k+1) ω))-
(fun _ => marchenkoPasturTransform t α)) 1 P =
ENNReal.ofReal (∫ ω, |gaussianGramStieltjes t (Z (k+1) ω)-marchenkoPasturTransform t α| ∂P) := by
change eLpNorm (fun ω => gaussianGramStieltjes t (Z (k+1) ω)-marchenkoPasturTransform t α) 1 P = _
rw [eLpNorm_one_eq_lintegral_enorm (hi k).aestronglyMeasurable,
← ofReal_integral_norm_eq_lintegral_enorm (hi k)]
simp only [Real.norm_eq_abs]
apply tendstoInMeasure_of_tendsto_eLpNorm (by norm_num : (1 : ℝ≥0∞) ≠ 0)
simpa only [he,ENNReal.ofReal_zero,Function.comp_def] using
(ENNReal.continuous_ofReal.tendsto 0).comp (gaussianPatternStieltjes_L1 hα ht P Z hZ)
end InvariantIsing
end
end OAI
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