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