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Journal Article

# On Extremal Theory for Stationary Processes

J. M. P. Albin
The Annals of Probability
Vol. 18, No. 1 (Jan., 1990), pp. 92-128
Stable URL: http://www.jstor.org/stable/2244229
Page Count: 37
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## Abstract

Let {ξ(t)}t ≥ 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 Rm-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 ε-upcrossings and local ε-maxima.

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