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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.
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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.
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The equivalence relation between random variables that differ only on a set of probability zero. More precisely, two random variables  $  X _ {1} $
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and  $  X _ {2} $,
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defined on a common probability space  $  ( \Omega , {\mathcal F} , {\mathsf P}) $,
 +
are called stochastically equivalent if  $  {\mathsf P} \{ X _ {1} = X _ {2} \} = 1 $.
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In most problems of probability theory one deals with classes of equivalent random variables, rather than with the random variables themselves.
  
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Two stochastic processes  $  X _ {1} ( t) $
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and  $  X _ {2} ( t) $,
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$  t \in T $,
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defined on a common probability space are called stochastically equivalent if for any  $  t \in T $
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stochastic equivalence holds between the corresponding random variables:  $  {\mathsf P} \{ X _ {1} ( t) = X _ {2} ( t) \} = 1 $.
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With regard to stochastic processes  $  X _ {1} ( t) $
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and  $  X _ {2} ( t) $
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with coinciding finite-dimensional distributions, the term "stochastic equivalence" is sometimes used in the broad sense.
  
 
====Comments====
 
====Comments====
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 {{MR|0731258}} {{ZBL|0549.31001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> 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) {{MR|0346882}} {{ZBL|0291.60019}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972) pp. 46 {{MR|0448504}} {{ZBL|0246.60032}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A.V. [A.V. Skorokhod] Skorohod, "Random processes with independent increments" , Kluwer (1991) pp. 9 (Translated from Russian) {{MR|1155400}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> R.Sh. Liptser, A.N. [A.N. Shiryaev] Shiryayev, "Theory of martingales" , Kluwer (1989) pp. 4 (Translated from Russian) {{MR|1022664}} {{ZBL|0728.60048}} </TD></TR></table>
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{|
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|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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|-
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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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|-
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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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|-
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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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|-
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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}}
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|}

Latest revision as of 08:23, 6 June 2020


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
How to Cite This Entry:
Stochastic equivalence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_equivalence&oldid=24298
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article