Open Access
Translator Disclaimer
January, 1990 On Extremal Theory for Stationary Processes
J. M. P. Albin
Ann. Probab. 18(1): 92-128 (January, 1990). DOI: 10.1214/aop/1176990940


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.


Download Citation

J. M. P. Albin. "On Extremal Theory for Stationary Processes." Ann. Probab. 18 (1) 92 - 128, January, 1990.


Published: January, 1990
First available in Project Euclid: 19 April 2007

zbMATH: 0704.60029
MathSciNet: MR1043939
Digital Object Identifier: 10.1214/aop/1176990940

Primary: 60G10
Secondary: 60G15 , 60G17 , 60G55

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

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 1 • January, 1990
Back to Top