| import Mathlib |
| import OAI.Probability.SKBarriers.Gaussian.GaussianCramer |
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|
| namespace OAI |
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| section |
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| noncomputable section |
| open scoped ENNReal NNReal Topology Interval |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem gaussianHeat_abs_le {f : ℝ → ℝ} {B : ℝ} (hb : ∀ x, |f x| ≤ B) |
| (a x : ℝ) : |gaussianHeat a f x| ≤ B := by |
| have H := norm_integral_le_of_norm_le (μ:=gaussianReal 0 1) (f:=fun z => f (x+a*z)) (integrable_const B) |
| (ae_of_all _ (fun z => by simpa only [Real.norm_eq_abs] using hb (x+a*z))) |
| simpa only [gaussianHeat,Real.norm_eq_abs,integral_const,probReal_univ,smul_eq_mul,one_mul] using H |
|
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| theorem gaussianHeat_continuous {f : ℝ → ℝ} (hf : Continuous f) |
| {B : ℝ} (hb : ∀ x, |f x| ≤ B) : |
| Continuous (fun p : ℝ × ℝ => gaussianHeat p.1 f p.2) := by |
| apply continuous_iff_continuousAt.mpr |
| intro p |
| apply continuousAt_of_dominated (bound:=fun _ : ℝ => B) |
| · exact Eventually.of_forall (fun a => (hf.comp (by fun_prop)).aestronglyMeasurable) |
| · exact Eventually.of_forall (fun a => ae_of_all _ (fun z => by |
| simpa only [Real.norm_eq_abs] using hb (a.2+a.1*z))) |
| · exact integrable_const _ |
| · exact ae_of_all _ (fun z => (hf.comp (by fun_prop)).continuousAt) |
|
|
| theorem gaussianHeat_variance_integral_lower {f df : ℝ → ℝ} |
| (hd : ∀ x, HasDerivAt f (df x) x) (hc : Continuous df) |
| {B C : ℝ} (hB : 0 ≤ B) (hC : 0 ≤ C) |
| (hb : ∀ x, |f x| ≤ B) (hdf : ∀ x, |df x| ≤ C) |
| (a b c : ℝ) (he : c^2=b^2+a^2) : |
| a^2*(∫ z, (gaussianHeat a df (b*z))^2 ∂gaussianReal 0 1) ≤ |
| (∫ z, f (c*z)^2 ∂gaussianReal 0 1)- |
| (∫ z, (gaussianHeat a f (b*z))^2 ∂gaussianReal 0 1) := by |
| have hf := continuous_iff_continuousAt.mpr (fun x => (hd x).continuousAt) |
| have hb2 (x : ℝ) : |f x^2| ≤ B^2 := by |
| rw [abs_of_nonneg (sq_nonneg _)] |
| simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hB).mpr (hb x) |
| have hdheat := gaussianHeat_continuous hc hdf |
| have hfheat := gaussianHeat_continuous hf hb |
| have hf2heat := gaussianHeat_continuous (hf.pow 2) hb2 |
| have hi1 : Integrable (fun z => (gaussianHeat a df (b*z))^2) (gaussianReal 0 1) := |
| gaussian_integrable_sq_bounded (hdheat.comp (show Continuous (fun z : ℝ => (a,b*z)) by fun_prop)) hC (fun z => gaussianHeat_abs_le hdf a _) |
| have hi2 : Integrable (fun z => gaussianHeat a (fun y => f y^2) (b*z)) (gaussianReal 0 1) := |
| gaussian_integrable_bounded (hf2heat.comp (show Continuous (fun z : ℝ => (a,b*z)) by fun_prop)) (fun z => gaussianHeat_abs_le hb2 a _) |
| have hi3 : Integrable (fun z => (gaussianHeat a f (b*z))^2) (gaussianReal 0 1) := |
| gaussian_integrable_sq_bounded (hfheat.comp (show Continuous (fun z : ℝ => (a,b*z)) by fun_prop)) hB (fun z => gaussianHeat_abs_le hb a _) |
| have H := integral_mono (hi1.const_mul (a^2)) (hi2.sub hi3) |
| (fun z => gaussianHeat_variance_lower hd hc hB hb hdf a (b*z)) |
| have HI := integral_sub hi2 hi3 |
| simp only [Pi.sub_apply] at H |
| rw [integral_const_mul,HI] at H |
| have HC := integral_gaussian_sum (hf.pow 2) |
| (HasExpGrowth.of_bounded (sq_nonneg B) (fun x => by |
| simpa only [Real.norm_eq_abs,Pi.pow_apply] using hb2 x)) b a c he |
| change (∫ z, gaussianHeat a (fun y => f y^2) (b*z) ∂gaussianReal 0 1)=_ at HC |
| rw [HC] at H |
| exact H |
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|
| theorem gaussianHeat_sq_average_continuous {f : ℝ → ℝ} (hf : Continuous f) |
| {B : ℝ} (hB : 0 ≤ B) (hb : ∀ x, |f x| ≤ B) : |
| Continuous (fun p : ℝ × ℝ => ∫ z, (gaussianHeat p.1 f (p.2*z))^2 ∂gaussianReal 0 1) := by |
| have hh := gaussianHeat_continuous hf hb |
| apply continuous_iff_continuousAt.mpr |
| intro p |
| apply continuousAt_of_dominated (bound:=fun _ : ℝ => B^2) |
| · exact Eventually.of_forall (fun a => ((hh.comp (show Continuous (fun z : ℝ => (a.1,a.2*z)) by fun_prop)).pow 2).aestronglyMeasurable) |
| · exact Eventually.of_forall (fun a => ae_of_all _ (fun z => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)] |
| simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hB).mpr (gaussianHeat_abs_le hb a.1 (a.2*z)))) |
| · exact integrable_const _ |
| · exact ae_of_all _ (fun z => ((hh.comp (show Continuous (fun a : ℝ × ℝ => (a.1,a.2*z)) by fun_prop)).pow 2).continuousAt) |
|
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| end SK.Analytic |
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| end |
| end |
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| end OAI |
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