Stochastic equivalence

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2010 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 and , defined on a common probability space , are called stochastically equivalent if . In most problems of probability theory one deals with classes of equivalent random variables, rather than with the random variables themselves.

Two stochastic processes and , , defined on a common probability space are called stochastically equivalent if for any stochastic equivalence holds between the corresponding random variables: . With regard to stochastic processes and with coinciding finite-dimensional distributions, the term "stochastic equivalence" is sometimes used in the broad sense.


The members of a stochastic equivalence class (of random variables or stochastic processes) are sometimes referred to as versions (of each other or of the equivalence class). A version of a random variable or stochastic process is also called a modification.


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How to Cite This Entry:
Stochastic equivalence. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article