Difference between revisions of "Stochastic continuity"
From Encyclopedia of Mathematics
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| − | + | ''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==== | |
| + | <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> | ||
| − | + | {{TEX|done}} | |
| − | |||
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=18160
Stochastic continuity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_continuity&oldid=18160
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article