Theorems · Theorem · probability

ProbabilityTheory.strong_law_ae

• #59 of the 100 theorems: The Laws of Large Numbers

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {E : Type u_2} [inst : NormedAddCommGroup E]
  [inst_1 : NormedSpace ℝ E] [CompleteSpace E] [inst_3 : MeasurableSpace E] [BorelSpace E] (X : ℕ → Ω → E),
  MeasureTheory.Integrable (X 0) μ →
    Pairwise (Function.onFun (fun x1 x2 => ProbabilityTheory.IndepFun x1 x2 μ) X) →
      (∀ (i : ℕ), ProbabilityTheory.IdentDistrib (X i) (X 0) μ μ) →
        ∀ᵐ (ω : Ω) ∂μ,
          Filter.Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, X i ω) Filter.atTop (nhds (∫ (x : Ω), X 0 x ∂μ))

Strong law of large numbers, almost sure version: if X n is a sequence of independent identically distributed integrable random variables taking values in a Banach space, then n⁻¹ • ∑ i ∈ range n, X i converges almost surely to 𝔼[X 0]. We give here the strong version, due to Etemadi, that only requires pairwise independence.

Defined in

Mathlib.Probability.StrongLaw

Cited by

1 results in Mathlib

Foundations

Depth 281 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpaceMeasurableSpaceBorelSpace