Difference between revisions of "Stochastic indistinguishability"
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| + | $#C+1 = 10 : ~/encyclopedia/old_files/data/S090/S.0900120 Stochastic indistinguishability | ||
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[[Category:Stochastic processes]] | [[Category:Stochastic processes]] | ||
| − | A property of two random processes | + | A property of two random processes $ X = ( X _ {t} ( \omega )) _ {t \geq 0 } $ |
| + | and $ Y = ( Y _ {t} ( \omega )) _ {t \geq 0 } $ | ||
| + | which states that the random set | ||
| − | + | $$ | |
| + | \{ X \neq Y \} = \ | ||
| + | \{ {( \omega , t) } : {X _ {t} ( \omega ) \neq Y _ {t} ( \omega ) } \} | ||
| + | $$ | ||
| − | can be disregarded, i.e. that the probability of the set | + | can be disregarded, i.e. that the probability of the set $ \{ \omega : {\exists t \geq 0 : ( \omega , t) \in \{ X \neq Y \} } \} $ |
| + | is equal to zero. If $ X $ | ||
| + | and $ Y $ | ||
| + | are stochastically indistinguishable, then $ X _ {t} = Y _ {t} $ | ||
| + | for all $ t \geq 0 $, | ||
| + | i.e. $ X $ | ||
| + | and $ Y $ | ||
| + | are stochastically equivalent (cf. [[Stochastic equivalence|Stochastic equivalence]]). The opposite, generally speaking, is not true, but for processes that are continuous from the right (left), stochastic indistinguishability follows from stochastic equivalence. | ||
====References==== | ====References==== | ||
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====Comments==== | ====Comments==== | ||
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====References==== | ====References==== | ||
Latest revision as of 08:23, 6 June 2020
2020 Mathematics Subject Classification: Primary: 60Gxx [MSN][ZBL]
A property of two random processes $ X = ( X _ {t} ( \omega )) _ {t \geq 0 } $ and $ Y = ( Y _ {t} ( \omega )) _ {t \geq 0 } $ which states that the random set
$$ \{ X \neq Y \} = \ \{ {( \omega , t) } : {X _ {t} ( \omega ) \neq Y _ {t} ( \omega ) } \} $$
can be disregarded, i.e. that the probability of the set $ \{ \omega : {\exists t \geq 0 : ( \omega , t) \in \{ X \neq Y \} } \} $ is equal to zero. If $ X $ and $ Y $ are stochastically indistinguishable, then $ X _ {t} = Y _ {t} $ for all $ t \geq 0 $, i.e. $ X $ and $ Y $ are stochastically equivalent (cf. Stochastic equivalence). The opposite, generally speaking, is not true, but for processes that are continuous from the right (left), stochastic indistinguishability follows from stochastic equivalence.
References
| [D] | C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972) MR0448504 Zbl 0246.60032 |
Comments
References
| [DM] | C. Dellacherie, P.A. Meyer, "Probabilities and potential" , A , North-Holland (1978) (Translated from French) MR0521810 Zbl 0494.60001 |
How to Cite This Entry:
Stochastic indistinguishability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_indistinguishability&oldid=26951
Stochastic indistinguishability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_indistinguishability&oldid=26951
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article