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{{MSC|60Gxx|60Axx}}

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====
{|
|valign="top"|{{Ref|Do}}|| J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 {{MR|0731258}} {{ZBL|0549.31001}}
|-
|valign="top"|{{Ref|GS}}|| I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , '''1''' , Springer (1974) pp. 43ff (Translated from Russian) {{MR|0346882}} {{ZBL|0291.60019}}
|-
|valign="top"|{{Ref|De}}|| C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972) pp. 46 {{MR|0448504}} {{ZBL|0246.60032}}
|-
|valign="top"|{{Ref|S}}|| A.V. Skorohod, "Random processes with independent increments" , Kluwer (1991) pp. 9 (Translated from Russian) {{MR|1155400}} {{ZBL|}}
|-
|valign="top"|{{Ref|LS}}|| R.Sh. Liptser, A.N. Shiryayev, "Theory of martingales" , Kluwer (1989) pp. 4 (Translated from Russian) {{MR|1022664}} {{ZBL|0728.60048}}
|}