## The Annals of Probability

- Ann. Probab.
- Volume 18, Number 1 (1990), 92-128.

### 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.

#### Article information

**Source**

Ann. Probab., Volume 18, Number 1 (1990), 92-128.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990940

**Digital Object Identifier**

doi:10.1214/aop/1176990940

**Mathematical Reviews number (MathSciNet)**

MR1043939

**Zentralblatt MATH identifier**

0704.60029

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G10: Stationary processes

Secondary: 60G15: Gaussian processes 60G17: Sample path properties 60G55: Point processes

**Keywords**

Extremal value theory crossings local maxima Poisson processes Gaussian processes Rayleigh processes star-shaped sets

#### Citation

Albin, J. M. P. On Extremal Theory for Stationary Processes. Ann. Probab. 18 (1990), no. 1, 92--128. doi:10.1214/aop/1176990940. https://projecteuclid.org/euclid.aop/1176990940