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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 47 48 49 50 51 52 53 54 55 56 | import OAI.Probability.InvariantIsing.Gaussian.GaussianGramTransformError
import OAI.Probability.InvariantIsing.Gaussian.GaussianPatternAspect
namespace OAI
/-! Convergence of the actual positive Gram transforms at every positive real argument. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped Topology
namespace InvariantIsing
lemma gaussianGramTransform_error_bound_tendsto {α t : ℝ} (hα : 0 ≤ α) (ht : 0 < t) :
Tendsto (fun k : ℕ =>
(((gaussianPatternCount α (k+1) : ℝ)/(k+1))*(Real.sqrt (2/((k+1 : ℝ)*t^2))+
1/((k+1 : ℝ)*t))+|(gaussianPatternCount α (k+1) : ℝ)/(k+1)-α|)/t)
atTop (𝓝 0) := by
have hn : Tendsto (fun k : ℕ => (k+1 : ℝ)) atTop atTop := by
simpa only [Function.comp_def,Nat.cast_add,Nat.cast_one] using
(tendsto_natCast_atTop_atTop (R := ℝ)).comp (tendsto_add_atTop_nat 1)
have hq : Tendsto (fun k : ℕ => (2 : ℝ)/((k+1 : ℝ)*t^2)) atTop (𝓝 0) :=
tendsto_const_nhds.div_atTop (hn.atTop_mul_const (sq_pos_of_pos ht))
have hr : Tendsto (fun k : ℕ => (1 : ℝ)/((k+1 : ℝ)*t)) atTop (𝓝 0) :=
tendsto_const_nhds.div_atTop (hn.atTop_mul_const ht)
have hs := (Real.continuous_sqrt.tendsto 0).comp hq
have ha := gaussianPatternCount_ratio_tendsto hα
simpa only [Function.comp_def,Real.sqrt_zero,mul_zero,add_zero,zero_add,sub_self,abs_zero,zero_div] using
((ha.mul (hs.add hr)).add ((ha.sub (tendsto_const_nhds (x := α))).abs)).div_const t
theorem gaussianGramTransform_mean_tendsto {α t : ℝ} (hα : 0 ≤ α) (ht : 0 < t) :
Tendsto (fun k : ℕ => ∫ z : EuclideanSpace ℝ (Fin (k+1) × Fin (gaussianPatternCount α (k+1))),
|gaussianGramStieltjes t z-marchenkoPasturTransform t α| ∂stdGaussian _) atTop (𝓝 0) := by
apply squeeze_zero (fun _ => integral_nonneg (fun _ => abs_nonneg _))
(fun k => ?_) (gaussianGramTransform_error_bound_tendsto hα ht)
simpa only [Nat.cast_add,Nat.cast_one] using gaussianGramTransform_mean_error
(N := k+1) (m := gaussianPatternCount α (k+1)) (Nat.succ_pos k) ht hα
theorem gaussianPatternStieltjes_L1 {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) :
Tendsto (fun k : ℕ => ∫ ω, |gaussianGramStieltjes t (Z (k+1) ω)-marchenkoPasturTransform t α| ∂P)
atTop (𝓝 0) := by
have he (k : ℕ) : (∫ ω, |gaussianGramStieltjes t (Z (k+1) ω)-marchenkoPasturTransform t α| ∂P) =
∫ z : EuclideanSpace ℝ (Fin (k+1) × Fin (gaussianPatternCount α (k+1))),
|gaussianGramStieltjes t z-marchenkoPasturTransform t α| ∂stdGaussian _ := by
simpa only [Function.comp_def,Pi.sub_apply] using (hZ (k+1)).integral_comp
(((continuous_gaussianGramStieltjes ht).sub continuous_const).abs.aestronglyMeasurable)
simp_rw [he]
exact gaussianGramTransform_mean_tendsto hα ht
end InvariantIsing
end
end OAI
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