Theorems · Theorem · probability

ProbabilityTheory.Kernel.HasSubgaussianMGF.integrable_exp_add_compProd

∀ {Ω : Type u_1} {Ω' : Type u_2} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} {ν : MeasureTheory.Measure Ω'}
  {κ : ProbabilityTheory.Kernel Ω' Ω} {X : Ω → ℝ} {c : NNReal} {Ω'' : Type u_3} {mΩ'' : MeasurableSpace Ω''}
  {Y : Ω'' → ℝ} {cY : NNReal} [MeasureTheory.SFinite ν] {η : ProbabilityTheory.Kernel (Ω' × Ω) Ω''}
  [ProbabilityTheory.IsZeroOrMarkovKernel η],
  ProbabilityTheory.Kernel.HasSubgaussianMGF X c κ ν →
    ProbabilityTheory.Kernel.HasSubgaussianMGF Y cY η (ν.compProd κ) →
      ∀ (t : ℝ), MeasureTheory.Integrable (fun ω => Real.exp (t * (X ω.1 + Y ω.2))) (ν.bind ⇑(κ.compProd η))

Defined in

Mathlib.Probability.Moments.SubGaussian

Cited by

1 results in Mathlib

Foundations

Depth 231 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.SFiniteProbabilityTheory.IsZeroOrMarkovKernel