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November, 1983 Sojourns of Stationary Processes in Rare Sets
Simeon M. Berman
Ann. Probab. 11(4): 847-866 (November, 1983). DOI: 10.1214/aop/1176993436


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.


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Simeon M. Berman. "Sojourns of Stationary Processes in Rare Sets." Ann. Probab. 11 (4) 847 - 866, November, 1983.


Published: November, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0562.60043
MathSciNet: MR714950
Digital Object Identifier: 10.1214/aop/1176993436

Primary: 60G10
Secondary: 60G15 , 60J60

Keywords: Gaussian process , limit distribution , Markov process , Sojourn , stationary process

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 4 • November, 1983
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