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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 | import OAI.Probability.InvariantIsing.Gaussian.GaussianGramResidualMean
namespace OAI
/-! Quantitative mean convergence of the actual Gram resolvent to the MP equation's root. -/
noncomputable section
open MeasureTheory ProbabilityTheory
namespace InvariantIsing
lemma gaussianGramTransform_error_integrable {N m : ℕ} {t : ℝ} (ht : 0 < t) (α : ℝ) :
Integrable (fun z : EuclideanSpace ℝ (Fin N × Fin m) =>
|gaussianGramStieltjes t z-marchenkoPasturTransform t α|) (stdGaussian _) := by
apply (integrable_const (1/t+|marchenkoPasturTransform t α|)).mono'
((continuous_gaussianGramStieltjes ht).sub continuous_const).abs.aestronglyMeasurable
apply ae_of_all
intro z
rw [Real.norm_eq_abs,abs_abs]
exact (abs_sub _ _).trans (add_le_add (by
rw [abs_of_nonneg (gaussianGramStieltjes_nonneg ht z)]
exact gaussianGramStieltjes_le ht z) le_rfl)
theorem gaussianGramTransform_mean_error {N m : ℕ} (hN : 0 < N) {t α : ℝ}
(ht : 0 < t) (hα : 0 ≤ α) :
(∫ z : EuclideanSpace ℝ (Fin N × Fin m),
|gaussianGramStieltjes t z-marchenkoPasturTransform t α| ∂stdGaussian _) ≤
(((m : ℝ)/N)*(Real.sqrt (2/((N : ℝ)*t^2))+1/((N : ℝ)*t))+|(m : ℝ)/N-α|)/t := by
have hi := gaussianGramResidual_abs_integrable (N := N) (m := m) ht
(show 0 ≤ (m : ℝ)/N by positivity)
calc
_ ≤ ∫ z : EuclideanSpace ℝ (Fin N × Fin m),
(|marchenkoPasturResidual t ((m : ℝ)/N) (gaussianGramStieltjes t z)|+|(m : ℝ)/N-α|)/t
∂stdGaussian _ := by
apply integral_mono (gaussianGramTransform_error_integrable ht α)
((hi.add (integrable_const _)).div_const t)
intro z
apply (marchenkoPasturTransform_error ht hα (gaussianGramStieltjes_nonneg ht z)).trans
apply div_le_div_of_nonneg_right _ ht.le
dsimp only [Pi.add_apply]
simpa only [abs_sub_comm α ((m : ℝ)/N)] using
marchenkoPasturResidual_coefficient (t := t) (α := α) (β := (m : ℝ)/N)
(gaussianGramStieltjes_nonneg ht z)
_ = ((∫ z : EuclideanSpace ℝ (Fin N × Fin m),
|marchenkoPasturResidual t ((m : ℝ)/N) (gaussianGramStieltjes t z)| ∂stdGaussian _)+
|(m : ℝ)/N-α|)/t := by
rw [integral_div,integral_add hi (integrable_const _)]
simp
_ ≤ _ := div_le_div_of_nonneg_right (add_le_add (gaussianGramResidual_mean_le hN ht) le_rfl) ht.le
end InvariantIsing
end
end OAI
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