| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.TiltedDirectionalMoment |
| import OAI.Probability.SKBarriers.Scalar.WindowIntegral |
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| namespace OAI |
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| section |
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| noncomputable section |
| open scoped BigOperators NNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
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| theorem fiberGaussian_linear_error_lipschitz (n : ℕ) (V F : ParameterSpace n → ℝ) |
| (hV : BoundedDerivs V) (hFc : ContDiff ℝ 1 F) (hFg : HasExpGrowth F) |
| (hFdg : HasExpGrowth (fderiv ℝ F)) (hFb : ∀ z, |F z|≤1) |
| {g : ℝ → ℝ} {M : ℝ≥0} (hg : LipschitzWith M g) (hb : ∀ y, |g y|≤1) |
| (L : ParameterSpace n →L[ℝ] ℝ) (x : ℝ) (a : Fin n → ℝ) |
| (hinv : L (coordinateVector n a)=0) (ε : ℝ) |
| (hscore : ∀ z, |fderiv ℝ V z (coordinateVector n a)|≤ε) : |
| |(∫ z, coordinateLinear n a z*F z*g (L z) ∂(fiberGaussian n x).tilted V)- |
| (∫ z, fderiv ℝ F z (coordinateVector n a)*g (L z) ∂(fiberGaussian n x).tilted V)|≤ε := by |
| let μ := (fiberGaussian n x).tilted V |
| let := fiberGaussian_tilted_probability n V hV x |
| let D : ParameterSpace n → ℝ := fun z => fderiv ℝ F z (coordinateVector n a) |
| have hDc : Continuous D := (hFc.continuous_fderiv (by norm_num)).clm_apply continuous_const |
| have hDg : HasExpGrowth D := hFdg.derivative_eval _ |
| have hDint : Integrable D μ := hDg.integrable_tilted_fiberGaussian n V hV hDc x |
| have hLFint : Integrable (fun z => coordinateLinear n a z*F z) μ := |
| ((HasExpGrowth.linear _).mul hFg).integrable_tilted_fiberGaussian n V hV |
| ((coordinateLinear n a).continuous.mul hFc.continuous) x |
| have Hlim₁ := scalarWindow_integral_tendsto hg hb L.measurable hLFint |
| have Hlim₂ := scalarWindow_integral_tendsto hg hb L.measurable hDint |
| have HI (k : ℕ) : |(∫ z, coordinateLinear n a z*F z*scalarWindow g (1/((k:ℝ)+1)) (L z) ∂μ)- |
| (∫ z, D z*scalarWindow g (1/((k:ℝ)+1)) (L z) ∂μ)|≤ε := by |
| let h : ℝ := 1/((k:ℝ)+1) |
| have hh : 0<h := by dsimp [h]; positivity |
| let G : ParameterSpace n → ℝ := fun z => scalarWindow g h (L z) |
| have hGc : ContDiff ℝ 1 G := (scalarWindow_contDiff hg.continuous h).comp L.contDiff |
| have hGgrow := scalarWindow_comp_growth hg.continuous (B:=1) hb hh L |
| have hGb (z) : |G z|≤1 := scalarWindow_bound hg.continuous (B:=1) hb hh (L z) |
| have hGi : TranslationInvariant G (coordinateVector n a) := by |
| intro z t |
| simp only [G,map_add,map_smul,hinv,smul_zero,add_zero] |
| have hd (z) : fderiv ℝ (fun z => F z*G z) z (coordinateVector n a)=D z*G z := by |
| rw [fderiv_fun_mul (hFc.differentiable (by norm_num) z) (hGc.differentiable (by norm_num) z)] |
| simp only [add_apply,smul_apply,smul_eq_mul, |
| hGi.fderiv_zero (hGc.differentiable (by norm_num)) z,mul_zero,zero_add,D,mul_comm] |
| have H := fiberGaussian_tilted_directional_stein n V (fun z => F z*G z) hV |
| (hFc.mul hGc) (hFg.mul hGgrow.1) |
| (HasExpGrowth.fderiv_mul (hFc.differentiable (by norm_num)) |
| (hGc.differentiable (by norm_num)) hFg hGgrow.1 hFdg hGgrow.2) x a |
| simp_rw [hd] at H |
| have he : (fun z => coordinateLinear n a z*(F z*G z))= |
| fun z => coordinateLinear n a z*F z*G z := by funext z; ring |
| rw [he] at H |
| change (∫ z, coordinateLinear n a z*F z*G z ∂μ)= |
| (∫ z, D z*G z ∂μ)+∫ z, F z*G z*fderiv ℝ V z (coordinateVector n a) ∂μ at H |
| change |(∫ z, coordinateLinear n a z*F z*G z ∂μ)-(∫ z, D z*G z ∂μ)|≤ε |
| rw [H,add_sub_cancel_left] |
| have Hb := norm_integral_le_of_norm_le_const (μ:=μ) |
| (f:=fun z => F z*G z*fderiv ℝ V z (coordinateVector n a)) |
| (C:=ε) (ae_of_all _ (fun z => by |
| rw [Real.norm_eq_abs,abs_mul,abs_mul] |
| have hFG : |F z| * |G z|≤1 := |
| (mul_le_mul (hFb z) (hGb z) (abs_nonneg _) zero_le_one).trans_eq (one_mul _) |
| exact (mul_le_mul hFG (hscore z) (abs_nonneg _) zero_le_one).trans_eq (one_mul _))) |
| simpa only [Real.norm_eq_abs,probReal_univ,mul_one] using Hb |
| exact le_of_tendsto (Hlim₁.sub Hlim₂).abs (Eventually.of_forall HI) |
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| end SK.Analytic |
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| end |
| end |
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| end OAI |
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