File size: 3,865 Bytes
135052a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | 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
|