Stochastic equivalence

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.

Comments

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.

References

[Do]	J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 MR0731258 Zbl 0549.31001
[GS]	I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , 1 , Springer (1974) pp. 43ff (Translated from Russian) MR0346882 Zbl 0291.60019
[De]	C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972) pp. 46 MR0448504 Zbl 0246.60032
[S]	A.V. Skorohod, "Random processes with independent increments" , Kluwer (1991) pp. 9 (Translated from Russian) MR1155400
[LS]	R.Sh. Liptser, A.N. Shiryayev, "Theory of martingales" , Kluwer (1989) pp. 4 (Translated from Russian) MR1022664 Zbl 0728.60048