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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 | import OAI.Probability.InvariantIsing.Gaussian.MPClippedTransform
namespace OAI
/-! Weak convergence in probability of the actual Gaussian empirical law after clipping. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped Topology
namespace InvariantIsing
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 // 0 < t}, TendstoInMeasure P
(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
end InvariantIsing
end
end OAI
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