On sequences of quasi-equivalent events II.

Révész, P. (1964) On sequences of quasi-equivalent events II. A MAGYAR TUDOMÁNYOS AKADÉMIA MATEMATIKAI KUTATÓ INTÉZETÉNEK KÖZLEMÉNYEI, 9 (1-2). pp. 227-233.

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Abstract

In [1] we have introduced the following: Definition. The events Aₙ defined on a probability space { Ω, δ, P } are called quasi-equivalent if the value of the ratio P(Aᵢ₁ Aᵢ₂ . . . Aᵢₖ/P(Aᵢ₁) P(Aᵢ₂) . . . P(Aᵢₖ) = αₖ (iⱼ ≠ iₗ if j ≠ l) (P(Aᵢ) > 0) depends only on k and it does not depend on the indices i₁, i₂, . . ., iₖ (k = 1, 2, . . .). The numbers α₁, α₂, . . . are called the moments of the quasiequivalent events A₁, A₂, . . . The paper [1] contains the characterization of infinite sequences of quasi-equivalent events under the restriction limₙ→∞ inf P(Aₙ) > 0. The main result of [1] can be summarized as follows: Theorem A. Let A₁, A₂, . . . be a sequence of quasi-equivalent events defined on the probability space { Ω, δ, P } such that limₙ→∞ inf P(Aₙ) > 0. Then there exists a random variable λ(ω) defined on { Ω, δ, P } with the following properties: (1) P{0 ≦ λ(ω) ≦ infₖ 1/P(Aₖ)} = 1 (2) M(λᵏ) = αₖ (k = 1, 2, . . .) where α₁, α₂, . . . are the moments of the events A₁, A₂, . . . (3) P(Aᵢ₁ Aᵢ₂ . . . Aᵢₖ ∣ λ) = P(Aᵢ₁ ∣ λ) P(Aᵢ₂ ∣ λ) . . . P(Aᵢₖ ∣ λ) = = λᵏ P(Aᵢ₁) P(Aᵢ₂) . . . P(Aᵢₖ) (with probability 1) (iⱼ ≠ iₗ if j ≠ l) (4) P{1/n ⁿ∑ₖ₌₁ αₖ(ω)/P(Aₖ) → λ(ω)} = 1 where αₖ(ω) is the indicator function of Aₖ, (5) ∞∏ₙ₌₁ β(Aₙ, Aₙ₊₁, . . .) = β(λ) where β(Aₙ, Aₙ₊₁, . . .) is the smallest σ-algebra which contains the events Aₙ, Aₙ₊₁, . . . and β(λ) is the smallest σ-algebra with respect to which λ(ω) is measurable. We say that two σ-algebras

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