openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Gaussian /GaussianGramTransformProbability.lean
| 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 | |