| import Mathlib | |
| import OAI.Probability.SKBarriers.Gaussian.GaussianAverageProduct | |
| import OAI.Probability.SKBarriers.Scalar.PathExponentialAverage | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open scoped BigOperators Topology | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| namespace SK.Analytic | |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace | |
| theorem hierarchyMomentLevel_exp_regular (n : ℕ) (m : Fin n → ℝ) | |
| {f g : ParameterSpace n → ℝ} (hf : BoundedDerivs f) | |
| (hg : Continuous g) (hG : HasExpGrowth g) (j : Fin (n+1)) : | |
| Continuous (hierarchyMomentLevel n m f g j) ∧ | |
| HasExpGrowth (hierarchyMomentLevel n m f g j) := by | |
| induction n with | |
| | zero => exact ⟨hg,hG⟩ | |
| | succ n ih => | |
| refine Fin.lastCases ?_ (fun j => ?_) j | |
| · simpa only [hierarchyMomentLevel,Fin.lastCases_last] using And.intro hg hG | |
| · have H := ih (fun i => m i.castSucc) (hf.gaussianStep (m (Fin.last n))) | |
| (gaussianAverage_continuous_of_expGrowth hf (m (Fin.last n)) hG hg) | |
| (gaussianAverage_expGrowth hf (m (Fin.last n)) hG) j | |
| simp only [hierarchyMomentLevel,Fin.lastCases_castSucc] | |
| refine ⟨H.1.comp continuous_fst,?_⟩ | |
| obtain ⟨C,M,hC,hM,hb⟩ := H.2 | |
| refine ⟨C,M,hC,hM,fun z => (hb z.1).trans ?_⟩ | |
| exact mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr | |
| (mul_le_mul_of_nonneg_left (norm_fst_le z) hM)) hC | |
| theorem hierarchyPathLaw_prefix_mul_exp (n : ℕ) (m : Fin n → ℝ) | |
| {f g h : ParameterSpace n → ℝ} (hf : BoundedDerivs f) | |
| (hg : Continuous g) (hh : Continuous h) (hG : HasExpGrowth g) (hH : HasExpGrowth h) | |
| (j : Fin (n+1)) (x : ℝ) : | |
| (∫ z, hierarchyMomentLevel n m f h j z*g z ∂hierarchyPathLaw n m f x) = | |
| ∫ z, hierarchyMomentLevel n m f h j z*hierarchyMomentLevel n m f g j z ∂hierarchyPathLaw n m f x := by | |
| have G := hierarchyMomentLevel_exp_regular n m hf hg hG j | |
| have H := hierarchyMomentLevel_exp_regular n m hf hh hH j | |
| rw [← hierarchyAverage_eq_integral_exp n m f hf (fun z => hierarchyMomentLevel n m f h j z*g z) (H.1.mul hg) (H.2.mul hG), | |
| ← hierarchyAverage_eq_integral_exp n m f hf (fun z => hierarchyMomentLevel n m f h j z*hierarchyMomentLevel n m f g j z) (H.1.mul G.1) (H.2.mul G.2)] | |
| exact hierarchyAverage_prefix_mul n m f g h hf j x | |
| theorem hierarchyPathLaw_fixed_mul_exp (n : ℕ) (m : Fin n → ℝ) | |
| {f g h : ParameterSpace n → ℝ} (hf : BoundedDerivs f) | |
| (hg : Continuous g) (hh : Continuous h) (hG : HasExpGrowth g) (hH : HasExpGrowth h) | |
| (j : Fin (n+1)) (hj : hierarchyMomentLevel n m f h j=h) (x : ℝ) : | |
| (∫ z, h z*g z ∂hierarchyPathLaw n m f x) = | |
| ∫ z, h z*hierarchyMomentLevel n m f g j z ∂hierarchyPathLaw n m f x := by | |
| have H := hierarchyPathLaw_prefix_mul_exp n m hf hg hh hG hH j x | |
| rwa [hj] at H | |
| theorem hierarchyMomentLevel_coordinate_fixed (n : ℕ) (m : Fin n → ℝ) | |
| {f : ParameterSpace n → ℝ} (hf : BoundedDerivs f) (i : Fin n) (j : Fin (n+1)) | |
| (hij : i.val < j.val) : | |
| hierarchyMomentLevel n m f (coordinateProjection n i) j=coordinateProjection n i := by | |
| induction n with | |
| | zero => exact Fin.elim0 i | |
| | succ n ih => | |
| revert hij | |
| refine Fin.lastCases (fun _ => ?_) (fun j hij => ?_) j | |
| · simp only [hierarchyMomentLevel,Fin.lastCases_last] | |
| · revert hij | |
| refine Fin.lastCases (fun hij => ?_) (fun i hij => ?_) i | |
| · simp only [Fin.val_last,Fin.val_castSucc] at hij | |
| omega | |
| · have he : (coordinateProjection (n+1) i.castSucc : ParameterSpace (n+1) → ℝ)= | |
| fun z => coordinateProjection n i z.1 := by funext z; exact coordinateProjection_castSucc n i z | |
| rw [he] | |
| simp only [hierarchyMomentLevel,Fin.lastCases_castSucc] | |
| rw [gaussianAverage_const_prefix hf,ih (fun i => m i.castSucc) (hf.gaussianStep _) i j hij] | |
| end SK.Analytic | |
| end | |
| end | |
| end OAI | |