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Difference between revisions of "Stochastic boundedness"

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''boundedness in probability''
 
''boundedness in probability''
  
The property of a [[Stochastic process|stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090030/s0900301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090030/s0900302.png" />, expressed by the condition: For an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090030/s0900303.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090030/s0900304.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090030/s0900305.png" />,
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The property of a [[stochastic process]] $X(t)$, $t \in \mathcal{T}$, expressed by the condition: For an arbitrary $\epsilon > 0$ there exists a $C > 0$ such that for all $t \in \mathcal{T}$,
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090030/s0900306.png" /></td> </tr></table>
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\mathbf{P}\{ |X(t)| > C \} < \epsilon \ .
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$$

Latest revision as of 14:48, 21 December 2014

boundedness in probability

The property of a stochastic process $X(t)$, $t \in \mathcal{T}$, expressed by the condition: For an arbitrary $\epsilon > 0$ there exists a $C > 0$ such that for all $t \in \mathcal{T}$, $$ \mathbf{P}\{ |X(t)| > C \} < \epsilon \ . $$

How to Cite This Entry:
Stochastic boundedness. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_boundedness&oldid=35772
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article