On Extremal Theory for Stationary Processes

Abstract

Let $\{\xi(t)\}_{t \geq 0}$ be a stationary stochastic process, with one-dimensional distribution function $G$. We develop a method to determine an asymptotic expression for $\Pr\{\sup_{0 \leq t \leq h} \xi(t) > u\}$, when $u \uparrow \sup\{v: G(v) < 1\}$, applicable when $G$ belongs to a domain of attraction of extremes, and we show that if $G$ belongs to such a domain, then so does the distribution function of $\sup_{0 \leq t \leq h} \xi(t)$. Applications are given to hitting probabilities for small sets for $\mathbb{R}^m$-valued Gaussian processes and to extrema of Rayleigh processes. Further, we prove the Gumbel, Frechet and Weibull laws, for maxima over increasing intervals, when $G$ is type I-, type II- and type III-attracted, respectively, and we establish the asymptotic Poisson character of $\varepsilon$-upcrossings and local $\varepsilon$-maxima.