Theorems · Theorem · probability

ProbabilityTheory.Kernel.HasSubgaussianMGF.add_comp

∀ {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''}
  [ProbabilityTheory.IsZeroOrMarkovKernel η],
  ProbabilityTheory.Kernel.HasSubgaussianMGF X c κ ν →
    ProbabilityTheory.Kernel.HasSubgaussianMGF Y cY η (ν.bind ⇑κ) →
      ProbabilityTheory.Kernel.HasSubgaussianMGF (fun p => X p.1 + Y p.2) (c + cY)
        (κ.compProd (ProbabilityTheory.Kernel.prodMkLeft Ω' η)) ν

For ν : Measure Ω', κ : Kernel Ω' Ω and η : Ω Ω'', if a random variable X : Ω → ℝ has a sub-Gaussian mgf with respect to κ and ν and another random variable Y : Ω'' → ℝ has a sub-Gaussian mgf with respect to η and κ ∘ₘ ν : Measure Ω, then X + Y (random variable on the measurable space Ω × Ω'') has a sub-Gaussian mgf with respect to κ ⊗ₖ prodMkLeft Ω' η : Kernel Ω' (Ω × Ω'') and ν.

Defined in

Mathlib.Probability.Moments.SubGaussian

Cited by

1 results in Mathlib

Foundations

Depth 296 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.SFiniteProbabilityTheory.IsZeroOrMarkovKernel