The Annals of Probability
- Ann. Probab.
- Volume 35, Number 2 (2007), 649-662.
0–1 laws for regular conditional distributions
Let (Ω, ℬ, P) be a probability space, a sub-σ-field, and μ a regular conditional distribution for P given . Necessary and sufficient conditions for μ(ω)(A) to be 0–1, for all and ω∈A0, where and P(A0)=1, are given. Such conditions apply, in particular, when is a tail sub-σ-field. Let H(ω) denote the -atom including the point ω∈Ω. Necessary and sufficient conditions for μ(ω)(H(ω)) to be 0–1, for all ω∈A0, are also given. If (Ω, ℬ) is a standard space, the latter 0–1 law is true for various classically interesting sub-σ-fields , including tail, symmetric, invariant, as well as some sub-σ-fields connected with continuous time processes.
Ann. Probab., Volume 35, Number 2 (2007), 649-662.
First available in Project Euclid: 30 March 2007
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Berti, Patrizia; Rigo, Pietro. 0–1 laws for regular conditional distributions. Ann. Probab. 35 (2007), no. 2, 649--662. doi:10.1214/009117906000000845. https://projecteuclid.org/euclid.aop/1175287757