| import Mathlib |
| import OAI.Probability.SKBarriers.Gaussian.GaussianAngle |
| import OAI.Probability.SKBarriers.Parisi.CDFHeatPrefix |
| import OAI.Probability.SKBarriers.Gaussian.C1Taylor |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped ENNReal NNReal Topology Interval |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem gaussian_covariance_derivative {f df : ℝ → ℝ} |
| (hd : ∀ x, HasDerivAt f (df x) x) (hc : Continuous df) |
| {B C : ℝ} (hb : ∀ x, |f x| ≤ B) (hdf : ∀ x, |df x| ≤ C) : |
| (∫ z, z*f z ∂gaussianReal 0 1)=∫ z, df z ∂gaussianReal 0 1 := by |
| have hf := continuous_iff_continuousAt.mpr (fun x => (hd x).continuousAt) |
| apply gaussian_integral_mul_eq_integral_deriv f df hd |
| (gaussian_integrable_bounded hf hb) (gaussian_integrable_bounded hc hdf) |
| apply ((gaussian_abs_pow_integrable 1).const_mul B).mono' |
| ((continuous_id.mul hf).aestronglyMeasurable) |
| exact ae_of_all _ (fun z => by |
| simp only [Pi.mul_apply,id_eq,Real.norm_eq_abs,abs_mul] |
| simpa only [pow_one,mul_comm] using mul_le_mul_of_nonneg_left (hb z) (abs_nonneg z)) |
|
|
| theorem gaussian_cramer_variance {f df : ℝ → ℝ} |
| (hd : ∀ x, HasDerivAt f (df x) x) (hc : Continuous df) |
| {B C : ℝ} (hB : 0 ≤ B) (hb : ∀ x, |f x| ≤ B) (hdf : ∀ x, |df x| ≤ C) : |
| (∫ z, df z ∂gaussianReal 0 1)^2 ≤ |
| (∫ z, f z^2 ∂gaussianReal 0 1)-(∫ z, f z ∂gaussianReal 0 1)^2 := by |
| let m := ∫ z, f z ∂gaussianReal 0 1 |
| let d := ∫ z, df z ∂gaussianReal 0 1 |
| have hf := continuous_iff_continuousAt.mpr (fun x => (hd x).continuousAt) |
| have hi := gaussian_integrable_bounded hf hb |
| have h2 := gaussian_integrable_sq_bounded hf hB hb |
| have hz : Integrable (fun z : ℝ => z) (gaussianReal 0 1) := (memLp_id_gaussianReal (μ:=0) (v:=1) 1).integrable le_rfl |
| have hz2 : Integrable (fun z : ℝ => z^2) (gaussianReal 0 1) := by |
| simpa only [sq_abs] using gaussian_abs_pow_integrable 2 |
| have hfz : Integrable (fun z : ℝ => z*f z) (gaussianReal 0 1) := by |
| apply ((gaussian_abs_pow_integrable 1).const_mul B).mono' |
| ((continuous_id.mul hf).aestronglyMeasurable) |
| exact ae_of_all _ (fun z => by |
| simp only [Pi.mul_apply,id_eq,Real.norm_eq_abs,abs_mul] |
| simpa only [pow_one,mul_comm] using mul_le_mul_of_nonneg_left (hb z) (abs_nonneg z)) |
| have heq (z : ℝ) : (f z-m-d*z)^2 = |
| f z^2-2*m*f z+m^2-2*d*(z*f z)+2*m*d*z+d^2*z^2 := by ring |
| have H : 0 ≤ ∫ z, (f z-m-d*z)^2 ∂gaussianReal 0 1 := integral_nonneg (fun _ => sq_nonneg _) |
| simp_rw [heq] at H |
| have hA := integral_sub h2 (hi.const_mul (2*m)) |
| have hB' := integral_add (h2.sub (hi.const_mul (2*m))) (integrable_const (m^2)) |
| have hC' := integral_sub ((h2.sub (hi.const_mul (2*m))).add (integrable_const (m^2))) |
| (hfz.const_mul (2*d)) |
| have hD' := integral_add (((h2.sub (hi.const_mul (2*m))).add (integrable_const (m^2))).sub |
| (hfz.const_mul (2*d))) (hz.const_mul (2*m*d)) |
| have hE' := integral_add ((((h2.sub (hi.const_mul (2*m))).add (integrable_const (m^2))).sub |
| (hfz.const_mul (2*d))).add (hz.const_mul (2*m*d))) (hz2.const_mul (d^2)) |
| simp only [Pi.add_apply,Pi.sub_apply] at hA hB' hC' hD' hE' |
| rw [hE',hD',hC',hB',hA] at H |
| rw [integral_const_mul (2*m*d) (fun z : ℝ => z), |
| show (∫ z : ℝ, z ∂gaussianReal 0 1)=0 from integral_id_gaussianReal] at H |
| simp only [integral_const_mul,integral_const,probReal_univ,smul_eq_mul,one_mul, |
| gaussian_second_moment,mul_zero,add_zero,mul_one, |
| gaussian_covariance_derivative hd hc hb hdf] at H |
| dsimp [m,d] at H ⊢ |
| nlinarith |
|
|
| theorem gaussianHeat_variance_lower {f df : ℝ → ℝ} |
| (hd : ∀ x, HasDerivAt f (df x) x) (hc : Continuous df) |
| {B C : ℝ} (hB : 0 ≤ B) (hb : ∀ x, |f x| ≤ B) (hdf : ∀ x, |df x| ≤ C) |
| (a x : ℝ) : a^2*(gaussianHeat a df x)^2 ≤ |
| gaussianHeat a (fun y => f y^2) x-(gaussianHeat a f x)^2 := by |
| have HD (z : ℝ) : HasDerivAt (fun z => f (x+a*z)) (a*df (x+a*z)) z := by |
| convert (hd (x+a*z)).comp z (((hasDerivAt_id z).const_mul a).const_add x) using 1 <;> |
| first | rfl | ring |
| have H := gaussian_cramer_variance HD (by fun_prop) hB (fun z => hb (x+a*z)) |
| (C:=|a| * C) (fun z => by |
| rw [abs_mul]; exact mul_le_mul_of_nonneg_left (hdf (x+a*z)) (abs_nonneg a)) |
| rw [integral_const_mul] at H |
| simpa only [mul_pow,gaussianHeat] using H |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
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|
