Stochastic equivalence

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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.


[a1] J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390
[a2] I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 1 , Springer (1974) pp. 43ff (Translated from Russian)
[a3] C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972) pp. 46
[a4] A.V. [A.V. Skorokhod] Skorohod, "Random processes with independent increments" , Kluwer (1991) pp. 9 (Translated from Russian)
[a5] R.Sh. Liptser, A.N. [A.N. Shiryaev] Shiryayev, "Theory of martingales" , Kluwer (1989) pp. 4 (Translated from Russian)
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