Sojourns of Stationary Processes in Rare Sets

Abstract

Let $X(t), t \geq 0$, be a stationary process assuming values in a measure space $B$. The family of measurable subsets $A_u, u > 0$ is called "rare" if $P(X(0) \in A_u) \rightarrow 0$ for $u \rightarrow \infty$. Put $L_t(u) = \operatorname{mes}\{s: 0 \leq s \leq t, X(s) \in A_u\}$. Under specified conditions it is shown that there exists a function $v = v(u)$ and a nonincreasing function $-\Gamma'(x)$ such that $P(v(u)L_t(u) > x)/E(v(u)L_t(u)) \rightarrow - \Gamma'(x), x > 0$, for $u \rightarrow \infty$ and fixed $t > 0$. If $u = u(t)$ varies appropriately with $t$, then, under suitable conditions, the random variable $v(u)L_t(u)$ has, for $t \rightarrow \infty$, a limiting distribution of the form of a compound Poisson distribution. The results are applied to Markov processes and Gaussian processes.