| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.ScalarTiltedExpectation |
|
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| namespace OAI |
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| section |
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| noncomputable section |
| open scoped Topology NNReal ContDiff |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem boundedDerivs_of_derivatives {f g dg : ℝ → ℝ} |
| (hg : ∀ x, HasDerivAt f (g x) x) (hdg : ∀ x, HasDerivAt g (dg x) x) |
| (hcont : Continuous dg) {B C : ℝ≥0} |
| (hB : ∀ x, |g x| ≤ B) (hC : ∀ x, |dg x| ≤ C) : BoundedDerivs f := by |
| have he : deriv f=g := funext (fun x => (hg x).deriv) |
| have he' : deriv g=dg := funext (fun x => (hdg x).deriv) |
| have hgc : ContDiff ℝ 1 g := contDiff_one_iff_deriv.mpr |
| ⟨fun x => (hdg x).differentiableAt,he' ▸ hcont⟩ |
| have hfc : ContDiff ℝ 2 f := by |
| rw [show (2:ℕ∞ω)=1+1 by norm_num,contDiff_succ_iff_deriv] |
| exact ⟨fun x => (hg x).differentiableAt,by norm_num,he ▸ hgc⟩ |
| have hfe : fderiv ℝ f=fun x => g x • (ContinuousLinearMap.id ℝ ℝ) := by |
| funext x |
| apply ContinuousLinearMap.ext |
| intro y |
| simp only [smul_apply,ContinuousLinearMap.id_apply,smul_eq_mul, |
| fderiv_eq_deriv_mul,he] |
| refine ⟨hfc,B,C,B.coe_nonneg,C.coe_nonneg,?_,?_⟩ |
| · intro x |
| rw [← norm_deriv_eq_norm_fderiv,he,Real.norm_eq_abs] |
| exact hB x |
| · intro x |
| rw [hfe,← norm_deriv_eq_norm_fderiv,((hdg x).smul_const (ContinuousLinearMap.id ℝ ℝ)).deriv, |
| norm_smul,ContinuousLinearMap.norm_id,Real.norm_eq_abs,mul_one] |
| exact hC x |
|
|
| def boundedPrimitive (g : ℝ → ℝ) (x : ℝ) : ℝ := ∫ y in 0..x, g y |
|
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| theorem boundedPrimitive_hasDerivAt {g : ℝ → ℝ} (hg : Continuous g) (x : ℝ) : |
| HasDerivAt (boundedPrimitive g) (g x) x := |
| intervalIntegral.integral_hasDerivAt_right (hg.intervalIntegrable 0 x) |
| hg.stronglyMeasurable.stronglyMeasurableAtFilter hg.continuousAt |
|
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| theorem boundedPrimitive_regular {g dg : ℝ → ℝ} |
| (hg : ∀ x, HasDerivAt g (dg x) x) (hdg : Continuous dg) |
| {B C : ℝ≥0} (hB : ∀ x, |g x| ≤ B) (hC : ∀ x, |dg x| ≤ C) : |
| BoundedDerivs (boundedPrimitive g) := |
| boundedDerivs_of_derivatives |
| (boundedPrimitive_hasDerivAt (continuous_iff_continuousAt.mpr (fun x => (hg x).continuousAt))) hg hdg hB hC |
|
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| theorem boundedPrimitive_lipschitz {g : ℝ → ℝ} (hg : Continuous g) {B : ℝ≥0} |
| (hB : ∀ x, |g x| ≤ B) : LipschitzWith B (boundedPrimitive g) := by |
| apply lipschitzWith_of_nnnorm_deriv_le (fun x => (boundedPrimitive_hasDerivAt hg x).differentiableAt) |
| intro x |
| rw [(boundedPrimitive_hasDerivAt hg x).deriv] |
| exact_mod_cast hB x |
|
|
| theorem dyadicScalarAverage_tendsto_C1 (β : ℝ) {α : ℝ → ℝ} |
| (hα : ∀ z, α z ∈ Icc (0:ℝ) 1) (hmono : Monotone α) |
| {f g dg : ℝ → ℝ} (hf : BoundedDerivs f) |
| (hg : ∀ y, HasDerivAt g (dg y) y) (hdg : Continuous dg) |
| {K B C : ℝ≥0} (hfK : LipschitzWith K f) |
| (hB : ∀ y, |g y| ≤ B) (hC : ∀ y, |dg y| ≤ C) |
| (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) (x : ℝ) : |
| Tendsto (fun n => dyadicScalarAverage β α n s t f g x) atTop |
| (𝓝 (scalarCDFAverage β α s t f g x)) := by |
| have hc : Continuous g := continuous_iff_continuousAt.mpr (fun y => (hg y).continuousAt) |
| exact dyadicScalarAverage_tendsto β hα hmono hf (boundedPrimitive_regular hg hdg hB hC) |
| (boundedPrimitive_hasDerivAt hc) hg hfK (boundedPrimitive_lipschitz hc hB) hB hC s t ht x |
|
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| end SK.Analytic |
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| end |
| end |
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| end OAI |
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