openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Gaussian /GaussianCompactVariance.lean
| import OAI.Probability.InvariantIsing.Gaussian.GaussianTestVariance | |
| import OAI.Probability.InvariantIsing.Gaussian.GaussianSmoothing | |
| import OAI.Probability.InvariantIsing.Pressure.VarianceLimit | |
| namespace OAI | |
| /-! Extending Gaussian Poincare from smooth to compact Lipschitz tests. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped NNReal Topology | |
| namespace InvariantIsing | |
| def gaussianApproxBump (d n : ℕ) : ContDiffBump (0 : EuclideanSpace ℝ (Fin d)) where | |
| rIn := (1/(n+1 : ℝ))/2 | |
| rOut := 1/(n+1 : ℝ) | |
| rIn_pos := by positivity | |
| rIn_lt_rOut := by | |
| have h : 0 < 1/(n+1 : ℝ) := by positivity | |
| linarith | |
| lemma gaussianApproxBump_rOut_le (d n : ℕ) : (gaussianApproxBump d n).rOut ≤ 1 := by | |
| change 1/(n+1 : ℝ) ≤ 1 | |
| apply (div_le_one (by positivity)).mpr | |
| linarith [Nat.cast_nonneg (α := ℝ) n] | |
| theorem gaussian_compact_variance_le {d : ℕ} {L : ℝ≥0} | |
| {f : EuclideanSpace ℝ (Fin d) → ℝ} (hf : LipschitzWith L f) | |
| (hc : HasCompactSupport f) : | |
| variance f (stdGaussian (EuclideanSpace ℝ (Fin d))) ≤ (L : ℝ)^2 := by | |
| let F := fun n => gaussianSmooth (gaussianApproxBump d n) f | |
| have hL n : LipschitzWith L (F n) := gaussianSmooth_lipschitz _ hf | |
| have hv n : variance (F n) (stdGaussian (EuclideanSpace ℝ (Fin d))) ≤ (L : ℝ)^2 := | |
| gaussian_test_variance_le (gaussianSmooth_test _ hf hc) (hL n) | |
| have ht (x : EuclideanSpace ℝ (Fin d)) : Tendsto (fun n => F n x) atTop (𝓝 (f x)) := by | |
| exact ContDiffBump.convolution_tendsto_right_of_continuous | |
| (show Tendsto (fun n => (gaussianApproxBump d n).rOut) atTop (𝓝 0) from | |
| tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)) hf.continuous x | |
| obtain ⟨B, hB⟩ := hc.exists_bound_of_continuous hf.continuous | |
| have hbound n x : ‖F n x‖ ≤ B+(L : ℝ) := by | |
| calc | |
| _ ≤ ‖f x‖+dist (F n x) (f x) := by | |
| simpa only [add_sub_cancel, dist_eq_norm] using norm_add_le (f x) (F n x-f x) | |
| _ ≤ B+(L : ℝ)*(gaussianApproxBump d n).rOut := | |
| add_le_add (hB x) (gaussianSmooth_dist_le _ hf x) | |
| _ ≤ B+(L : ℝ) := by | |
| nlinarith [mul_le_mul_of_nonneg_left (gaussianApproxBump_rOut_le d n) L.coe_nonneg] | |
| have hm : Tendsto (fun n => ∫ x, F n x ∂stdGaussian _) atTop | |
| (𝓝 (∫ x, f x ∂stdGaussian _)) := by | |
| apply tendsto_integral_of_dominated_convergence (fun _ => B+(L : ℝ)) | |
| (fun n => (hL n).continuous.aestronglyMeasurable) (integrable_const _) | |
| · exact fun n => ae_of_all _ (hbound n) | |
| · exact ae_of_all _ ht | |
| exact variance_le_of_ae_tendsto_of_mean_tendsto | |
| (fun n => gaussian_lipschitz_memLp_two (hL n)) (gaussian_lipschitz_memLp_two hf) | |
| (ae_of_all _ ht) hm hv | |
| end InvariantIsing | |
| end | |
| end OAI | |