The Annals of Probability

Extremal Theory for Stochastic Processes

M. R. Leadbetter and Holger Rootzen

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The purpose of this paper is to provide an overview of the asymptotic distributional theory of extreme values for a wide class of dependent stochastic sequences and continuous parameter processes. The theory contains the standard classical extreme value results for maxima and extreme order statistics as special cases but is richer on account of the diverse behavior possible under dependence in both discrete and continuous time contexts. Emphasis is placed on stationary cases but some departures from stationarity are considered. Significant ideas and methods are described rather than details, and, in particular, the nature and role of important underlying point processes (such as exceedances and upcrossings) are emphasized. Applications are given to particular classes of processes (e.g., normal, moving average) and connections with related theory (such as convergence of sums) are indicated.

Article information

Ann. Probab., Volume 16, Number 2 (1988), 431-478.

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 60F05: Central limit and other weak theorems

60-02 Extreme values stationary processes point processes


Leadbetter, M. R.; Rootzen, Holger. Extremal Theory for Stochastic Processes. Ann. Probab. 16 (1988), no. 2, 431--478. doi:10.1214/aop/1176991767.

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