# Stochastic equivalence

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2020 Mathematics Subject Classification: Primary: 60Gxx Secondary: 60Axx [MSN][ZBL]

The equivalence relation between random variables that differ only on a set of probability zero. More precisely, two random variables $X _ {1}$ and $X _ {2}$, defined on a common probability space $( \Omega , {\mathcal F} , {\mathsf P})$, are called stochastically equivalent if ${\mathsf P} \{ X _ {1} = X _ {2} \} = 1$. In most problems of probability theory one deals with classes of equivalent random variables, rather than with the random variables themselves.

Two stochastic processes $X _ {1} ( t)$ and $X _ {2} ( t)$, $t \in T$, defined on a common probability space are called stochastically equivalent if for any $t \in T$ stochastic equivalence holds between the corresponding random variables: ${\mathsf P} \{ X _ {1} ( t) = X _ {2} ( t) \} = 1$. With regard to stochastic processes $X _ {1} ( t)$ and $X _ {2} ( t)$ with coinciding finite-dimensional distributions, the term "stochastic equivalence" is sometimes used in the broad sense.