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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 | import Mathlib
import OAI.Probability.SKBarriers.Calculus.ParameterLinear
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter
namespace SK.Analytic
section ParameterInverse
variable {P E : Type} [NormedAddCommGroup P] [NormedAddCommGroup E]
theorem ParamLinearGrowth.inv_gaussian_exp {f : P × (E × ℝ) → ℝ}
(hf : ParamLinearGrowth f) (hc : Continuous f) :
ParamExpGrowth (fun z : P × E => (∫ y, Real.exp (f (z.1,(z.2,y))) ∂gaussianReal 0 1)⁻¹) := by
intro R
obtain ⟨C,hC,hf⟩ := hf R
let B := ∫ y : ℝ, Real.exp (-C*|y|) ∂gaussianReal 0 1
have hB : 0 < B := integral_exp_pos (integrable_exp_mul_abs_gaussian (-C))
refine ⟨Real.exp C/B,C,by positivity,hC,?_⟩
intro p e he
let A := C*(1+‖e‖)
have hb y : |f (p,(e,y))| ≤ A+C*|y| := by
apply (hf p (e,y) he).trans
rw [Prod.norm_def,Real.norm_eq_abs]
have hh := max_le_add_of_nonneg (norm_nonneg e) (abs_nonneg y)
dsimp [A]
nlinarith
have H := gaussian_exp_integral_bounds (fun y => f (p,(e,y)))
(hc.comp (continuous_const.prodMk (continuous_const.prodMk continuous_id))) hC hb
let Z := ∫ y, Real.exp (f (p,(e,y))) ∂gaussianReal 0 1
have hlo : 0 < Real.exp (-A)*B := mul_pos (Real.exp_pos _) hB
have hz : 0 < Z := hlo.trans_le H.1
change ‖Z⁻¹‖ ≤ Real.exp C/B*Real.exp (C*‖e‖)
rw [Real.norm_eq_abs,abs_of_pos (inv_pos.2 hz)]
calc
Z⁻¹ ≤ (Real.exp (-A)*B)⁻¹ := (inv_le_inv₀ hz hlo).2 H.1
_ = Real.exp C/B*Real.exp (C*‖e‖) := by
rw [mul_inv_rev,Real.exp_neg,inv_inv]
dsimp [A]
rw [mul_add,mul_one,Real.exp_add]
ring
end ParameterInverse
end SK.Analytic
end
end
end
end OAI
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