Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-14T15:16:14.215Z Has data issue: false hasContentIssue false
  • English
  • Français

Compound Poisson approximation for the distribution of extremes

Part of: Stochastic processes Limit theorems

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

Arratia, R., Goldstein, L. and Gordon, L. (1990). Poisson approximation and the Chen–Stein method. Statist. Sci. 5, 403–434.Google Scholar
Barbour, A. D. and Brown, T. C. (1992). Stein's method and point process approximation. Stoch. Process. Appl. 43, 9–31.Google Scholar
Barbour, A. D., Gerrard, R. and Reinert, G. (2000). Iterates of expanding maps. Prob. Theory Relat. Fields 116, 151–180.Google Scholar
Berbee, H. C. P. (1979). Random Walks with Stationary Increments and Renewal Theory (Math. Centre Tracts 112). Mathematisch Centrum, Amsterdam.Google Scholar
Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth and Brooks/Cole, Pacific Grove, CA.CrossRefGoogle Scholar
Bosq, D. (1996). Nonparametric Statistics for Stochastic Processes. Springer, Berlin.Google Scholar
Brown, T. C. and Xia, A. (1995). On metrics in point process approximation. Stoch. Stoch. Rep. 52, 247–263.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.Google Scholar
Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. John Wiley, New York.Google Scholar
Hsing, T. (1993). On some estimates based on sample behaviour near high level excursions. Prob. Theory Relat. Fields 95, 331–356.Google Scholar
Hsing, T., Hüsler, J. and Leadbetter, M. R. (1988). On the exceedance point process for a stationary sequence. Prob. Theory Relat. Fields 78, 97–112.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.Google Scholar
Lindeberg, Y. W. (1922). Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung. Math. Z. 15, 221–225.Google Scholar
Novak, S. Y. (1996). Extreme values in stationary sequences. Siberian Adv. Math. 6, 68–80.Google Scholar
Novak, S. Y. (1998). On the limiting distribution of extremes. Siberian Adv. Math. 8, 70–95.Google Scholar
O'Brien, G. (1974). Limit theorems for the maximum term of a stationary process. Ann. Prob. 2, 540–545.CrossRefGoogle Scholar
Xia, A. (1997). On the rate of Poisson process approximation to a Bernoulli process. J. Appl. Prob. 34, 898–907.Google Scholar