| import Mathlib |
| import OAI.Probability.SKBarriers.Calculus.ParameterGrowth |
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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 ParameterWeakGrowth |
| variable {P E F : Type} [NormedAddCommGroup P] [NormedAddCommGroup E] [Norm F] |
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| theorem ParamExpGrowth.weakNorm {f : P × E → F} (hf : ParamExpGrowth f) |
| (hn : ∀ z, 0 ≤ ‖f z‖) : ParamExpGrowth (fun z => ‖f z‖) := by |
| apply hf.of_norm_le zero_le_one |
| intro z |
| rw [Real.norm_eq_abs,abs_of_nonneg (hn z),one_mul] |
|
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| theorem ParamExpGrowth.weakLocallyDominated {f : P × (E × ℝ) → F} |
| (hf : ParamExpGrowth f) : |
| GaussianLocallyDominated (fun z : (P × E) × ℝ => f (z.1.1,(z.1.2,z.2))) := by |
| intro R |
| obtain ⟨C,M,hC,hM,hf⟩ := hf R |
| refine ⟨fun y => C*Real.exp (M*R)*Real.exp (M*|y|), |
| (integrable_exp_mul_abs_gaussian M).const_mul _,?_⟩ |
| rintro ⟨p,e⟩ y hpe |
| have hp : ‖p‖ ≤ R := (le_max_left _ _).trans hpe |
| have he : ‖e‖ ≤ R := (le_max_right _ _).trans hpe |
| apply (hf p (e,y) hp).trans |
| change C*Real.exp (M*‖(e,y)‖) ≤ C*Real.exp (M*R)*Real.exp (M*|y|) |
| rw [mul_assoc,← Real.exp_add,← mul_add] |
| apply mul_le_mul_of_nonneg_left _ hC |
| apply Real.exp_le_exp.2 |
| apply mul_le_mul_of_nonneg_left _ hM |
| rw [Prod.norm_def,Real.norm_eq_abs] |
| exact max_le (he.trans (le_add_of_nonneg_right (abs_nonneg _))) |
| (le_add_of_nonneg_left ((norm_nonneg e).trans he)) |
| end ParameterWeakGrowth |
| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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