| import Mathlib |
| import OAI.Probability.SKBarriers.Calculus.ParameterLinear |
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|
| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| namespace SK.Analytic |
| section ParameterInverse |
| variable {P E : Type} [NormedAddCommGroup P] [NormedAddCommGroup E] |
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| 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 |
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| end |
| end |
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| end |
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| end OAI |
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