Let $(Omega,mathcal{B},P)$ be a probability space, $mathcal{A}subsetmathcal{B}$ a sub-$sigma$-field, and $mu$ a regular conditional distribution for $P$ given $mathcal{A}$. Necessary and sufficient conditions for $mu(omega)(A)$ to be 0-1, for all $Ainmathcal{A}$ and $omegain A_0$, where $A_0inmathcal{A}$ and $P(A_0)=1$, are given. Such conditions apply, in particular, when $mathcal{A}$ is a tail sub-$sigma$-field. Let $H(omega)$ denote the $mathcal{A}$-atom including the point $omegainOmega$. Necessary and sufficient conditions for $mu(omega)(H(omega))$ to be 0-1, for all $omegain A_0$, are also given. If $(Omega,mathcal{B})$ is a standard space, the latter 0-1 law is true for various classically interesting sub-$sigma$-fields $mathcal{A}$, including tail, symmetric, invariant, as well as some sub-$sigma$-fields connected with continuous time processes.

Patrizia Berti, Pietro Rigo (2007). 0-1 laws for regular conditional distributions. ANNALS OF PROBABILITY, 35, 649-662 [10.1214/009117906000000845].

0-1 laws for regular conditional distributions

Pietro Rigo
2007

Abstract

Let $(Omega,mathcal{B},P)$ be a probability space, $mathcal{A}subsetmathcal{B}$ a sub-$sigma$-field, and $mu$ a regular conditional distribution for $P$ given $mathcal{A}$. Necessary and sufficient conditions for $mu(omega)(A)$ to be 0-1, for all $Ainmathcal{A}$ and $omegain A_0$, where $A_0inmathcal{A}$ and $P(A_0)=1$, are given. Such conditions apply, in particular, when $mathcal{A}$ is a tail sub-$sigma$-field. Let $H(omega)$ denote the $mathcal{A}$-atom including the point $omegainOmega$. Necessary and sufficient conditions for $mu(omega)(H(omega))$ to be 0-1, for all $omegain A_0$, are also given. If $(Omega,mathcal{B})$ is a standard space, the latter 0-1 law is true for various classically interesting sub-$sigma$-fields $mathcal{A}$, including tail, symmetric, invariant, as well as some sub-$sigma$-fields connected with continuous time processes.
2007
Patrizia Berti, Pietro Rigo (2007). 0-1 laws for regular conditional distributions. ANNALS OF PROBABILITY, 35, 649-662 [10.1214/009117906000000845].
Patrizia Berti; Pietro Rigo
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/738718
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