View source for Stochastic equivalence

{{MSC|60Gxx|60Axx}}

The equivalence relation between random variables that differ only on a set of probability zero. More precisely, two random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090090/s0900901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090090/s0900902.png" />, defined on a common probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090090/s0900903.png" />, are called stochastically equivalent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090090/s0900904.png" />. In most problems of probability theory one deals with classes of equivalent random variables, rather than with the random variables themselves.

Two stochastic processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090090/s0900905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090090/s0900906.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090090/s0900907.png" />, defined on a common probability space are called stochastically equivalent if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090090/s0900908.png" /> stochastic equivalence holds between the corresponding random variables: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090090/s0900909.png" />. With regard to stochastic processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090090/s09009010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090090/s09009011.png" /> 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}}
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|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}}
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|valign="top"|{{Ref|De}}|| C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972) pp. 46 {{MR|0448504}} {{ZBL|0246.60032}}
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|valign="top"|{{Ref|S}}|| A.V. Skorohod, "Random processes with independent increments" , Kluwer (1991) pp. 9 (Translated from Russian) {{MR|1155400}} {{ZBL|}}
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|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}}
|}