| import OAI.Probability.InvariantIsing.Gaussian.MPClippedTransform |
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| namespace OAI |
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| /-! Weak convergence in probability of the actual Gaussian empirical law after clipping. -/ |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped Topology |
| namespace InvariantIsing |
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| theorem gaussianPattern_clipped_weak {α : ℝ} (hα : 0 < α) |
| {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] |
| (Z : (N : ℕ) → Ω → EuclideanSpace ℝ (Fin N × Fin (gaussianPatternCount α N))) |
| (hZ : ∀ N, HasLaw (Z N) (stdGaussian _) P) : |
| TendstoInMeasure P (fun k ω => LevyProkhorov.ofMeasure |
| ((empiricalSpectralLaw (Nat.succ_pos k) (gaussianPatternEigenvalues (Z (k+1) ω))).map |
| (spectralClip 0 (marchenkoPasturB α)))) atTop |
| (fun _ => LevyProkhorov.ofMeasure (⟨marchenkoPasturMeasure α,marchenkoPastur_probability hα⟩ : ProbabilityMeasure ℝ)) := by |
| let R := marchenkoPasturB α |
| have hR : 0 ≤ R := sq_nonneg _ |
| let ν : ProbabilityMeasure ℝ := ⟨marchenkoPasturMeasure α,marchenkoPastur_probability hα⟩ |
| let W := fun k ω => compactClippedLaw R hR |
| (empiricalSpectralLaw (Nat.succ_pos k) (gaussianPatternEigenvalues (Z (k+1) ω))) |
| have hh : ∀ t : {t : ℝ |
| (fun k ω => ∫ x, compactResolventTest R t x ∂(W k ω : Measure _)) atTop |
| (fun _ => ∫ x, compactResolventTest R t x ∂(compactClippedLaw R hR ν : Measure _)) := by |
| intro t |
| simp only [W,compactClippedLaw_integral] |
| change TendstoInMeasure P _ atTop (fun _ => ∫ x, |
| positiveResolventTest t (spectralClip 0 (marchenkoPasturB α) x) ∂marchenkoPasturMeasure α) |
| rw [marchenkoPastur_clipped_transform hα t.property] |
| exact gaussianPattern_clipped_stieltjes hα t.property P Z hZ |
| have hc := compact_resolvent_tendstoInMeasure R P W (compactClippedLaw R hR ν) hh |
| have hn : (compactClippedLaw R hR ν).map (fun x => (x : ℝ)) = ν := |
| compactClippedLaw_real_eq R hR ν (marchenkoPastur_mem_nonnegative_interval hα) |
| rw [hn] at hc |
| simpa only [W,compactClippedLaw_real] using hc |
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| end InvariantIsing |
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| end |
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| end OAI |
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