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

Berman, Simeon M.

Ann. Probab. 11, 847-866 (1983).

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

Keywords:

extreme values; asymptotic behaviour; sojourn time; local sojourn theorem; mixing condition

Citations:

Zbl 0498.60035; Zbl 0518.60021