Stochastic Continuity

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 $X(t)$.

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