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Functionals of clusters of extremes

Part of: Stochastic processes Nonparametric inference

Published online by Cambridge University Press:  01 July 2016

Johan Segers
Johan Segers*
Affiliation:
Tilburg University and Catholic University Leuven
*
∗ Postal address: Tilburg University, PO Box 90153, NL-5000 LE Tilburg, The Netherlands. Email address: jsegers@uvt.nl
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Abstract

For arbitrary stationary sequences of random variables satisfying a mild mixing condition, distributional approximations are established for functionals of clusters of exceedances over a high threshold. The approximations are in terms of the distribution of the process conditionally on the event that the first variable exceeds the threshold. This conditional distribution is shown to converge to a nontrivial limit if the finite-dimensional distributions of the process are in the domain of attraction of a multivariate extreme-value distribution. In this case, therefore, limit distributions are obtained for functionals of clusters of extremes, thereby generalizing results for higher-order stationary Markov chains by Yun (2000).

Keywords

Cluster functionalextremal indexextreme-value distributionstationary sequencetail sequencethreshold exceedance

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62G32: Statistics of extreme values; tail inference
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 4 , December 2003 , pp. 1028 - 1045
Copyright
Copyright © Applied Probability Trust 2003 

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