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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 57 58 59 60 61 62 | 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
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