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Compound Poisson approximation for the distribution of extremes

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

A. D. Barbour ,
S. Y. Novak  and
A. Xia
A. D. Barbour*
Affiliation:
Universität Zürich
S. Y. Novak
Affiliation:
EURANDOM, Eindhoven
A. Xia
Affiliation:
University of New South Wales
*
∗ Postal address: Angewandte Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland. Email address: adb@amath.unizh.ch
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Abstract

Empirical point processes of exceedances play an important role in extreme value theory, and their limiting behaviour has been extensively studied. Here, we provide explicit bounds on the accuracy of approximating an exceedance process by a compound Poisson or Poisson cluster process, in terms of a Wasserstein metric that is generally more suitable for the purpose than the total variation metric. The bounds only involve properties of the finite, empirical sequence that is under consideration, and not of any limiting process. The argument uses Bernstein blocks and Lindeberg's method of compositions.

Keywords

Extreme valuespoint processesexceedancescompound Poisson processtotal variation metriccoupling

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 1 , March 2002 , pp. 223 - 240
Copyright
Copyright © Applied Probability Trust 2002 

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Footnotes

∗∗

Current address: Brunel University, Uxbridge, Middlesex UB8 3PH, UK.

∗∗∗

Current address: Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia.

Supported in part by Schweizer Nationalfonds Projekt Nr. 20-50686.97.

Supported by an Australian Research Council Small Grant from the University of New South Wales.

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