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Difference between revisions of "Stochastic continuity"

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A property of the sample functions of a [[Stochastic process|stochastic process]]. A stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090040/s0900401.png" /> defined on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090040/s0900402.png" /> is called stochastically continuous on this set if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090040/s0900403.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090040/s0900404.png" />,
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''continuity in probability''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090040/s0900405.png" /></td> </tr></table>
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A property of the sample functions of a [[stochastic process]]. A stochastic process $X(t)$ defined on a set $T \subseteq \mathbf{R}^1$ is called stochastically continuous on this set if for any $\epsilon > 0$ and all $t_0$,
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$$
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\lim_{t \rightarrow t_0} \mathbf{P}\{\rho(X(t),X(t_0)) > \epsilon\} = 0
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$$
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where $\rho$ is the distance between points in the corresponding space of values of $X(t)$.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090040/s0900406.png" /> is the distance between points in the corresponding space of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090040/s0900407.png" />.
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====References====
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. [Yu.V. Prokhorov] Prohorov,  Yu.A. Rozanov,  "Probability theory, basic concepts. Limit theorems, random processes" , Springer  (1979)  (Translated from Russian)</TD></TR>
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</table>
  
====References====
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{{TEX|done}}
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. [Yu.V. Prokhorov] Prohorov,  Yu.A. Rozanov,  "Probability theory, basic concepts. Limit theorems, random processes" , Springer  (1979)  (Translated from Russian)</TD></TR></table>
 

Latest revision as of 19:52, 2 November 2016

continuity in probability

A property of the sample functions of a stochastic process. A stochastic process $X(t)$ defined on a set $T \subseteq \mathbf{R}^1$ is called stochastically continuous on this set if for any $\epsilon > 0$ and all $t_0$, $$ \lim_{t \rightarrow t_0} \mathbf{P}\{\rho(X(t),X(t_0)) > \epsilon\} = 0 $$ where $\rho$ is the distance between points in the corresponding space of values of $X(t)$.

References

[1] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1979) (Translated from Russian)
How to Cite This Entry:
Stochastic continuity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_continuity&oldid=39602
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article