## Sojourns of stationary processes in rare sets.(English)Zbl 0562.60043

Let X(t), $$t\geq 0$$, be a separable, measurable stationary (vector) process. A family of measurable sets $$A_ u$$, $$u>0$$, is called rare, if $$P(X(0)\in A_ u)\to 0$$ as $$u\to \infty$$, e.g., in the theory of extreme values of real valued processes $$A_ u=(u,\infty)$$. The author presents generalizations of his earlier results on the asymptotic behaviour of the sojourn time of X(s), $$0\leq s\leq t$$, in $$A_ u$$, $$L_ t(u)=\int^{t}_{0}Ind_{\{X(s)\in A_ u\}}ds.$$
In fact, a local sojourn theorem presented by the author in ibid. 10, 1- 46 (1982; Zbl 0498.60035) is generalized and it is shown that under specified conditions there exists a function v and a non-increasing function -$$\Gamma$$ ’ such that $P(v(u)L_ t(u)>x)/E(v(u)L_ t(u))\to -\Gamma '(x),$ x$$>0$$, for $$u\to \infty$$ and fixed $$t>0.$$
The second main result is a global sojourn theorem stating that $$v(u)L_ t(u)$$ is asymptotically compound Poisson distributed under a mixing condition on the family $$A_ u$$ similar to the mixing condition of M. R. Leadbetter, G. Lindgren and H. Rootzén, Extremes and related properties of random sequences and processes. (1983; Zbl 0518.60021).
The results are applied to Markov processes and multivariate Gaussian processes.
Reviewer: H.Niemi

### MSC:

 60G10 Stationary stochastic processes 60G15 Gaussian processes 60J60 Diffusion processes

### Citations:

Zbl 0498.60035; Zbl 0518.60021
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