The Annals of Probability

On Extremal Theory for Stationary Processes

J. M. P. Albin

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G10: Stationary processes
Secondary: 60G15: Gaussian processes 60G17: Sample path properties 60G55: Point processes

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


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

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