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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: [email protected]
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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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References

Bortot, P. and Tawn, J. A. (1998). Models for the extremes of Markov chains. Biometrika 85, 851–867.Google Scholar
De Haan, L. (1984). A spectral representation for max-stable processes. Ann. Prob. 12, 1194–1204.Google Scholar
De Haan, L. and Pickands, J. (1986). Stationary min-stable stochastic processes. Prob. Theory Relat. Fields 72, 477–492.Google Scholar
Leadbetter, M. R. (1983). Extremes and local dependence of stationary sequences. Z. Wahrscheinlichkeitsth. 65, 291–306.Google Scholar
Leadbetter, M. R. (1991). On a basis for ‘peaks over threshold’ modeling. Statist. Prob. Lett. 12, 357–362.CrossRefGoogle Scholar
Ledford, A. W. and Tawn, J. A. (1997). Modelling dependence within joint tail regions. J. R. Statist. Soc. B 59, 475–499.Google Scholar
Ledford, A. W. and Tawn, J. A. (2003). Diagnostics for dependence within time series extremes. J. R. Statist. Soc. B 65, 521–543.Google Scholar
Marshall, A. W. and Olkin, I. (1983). Domains of attraction of multivariate extreme value distributions. Ann. Prob. 11, 168–177.Google Scholar
O'Brien, G. L. (1987). Extreme values for stationary and Markov sequences. Ann. Prob. 15, 281–291.Google Scholar
Perfekt, R. (1994). Extremal behaviour of stationary Markov chains with applications. Ann. Appl. Prob. 4, 529–548.CrossRefGoogle Scholar
Pickands, J. (1975). Statistical inference using extreme order statistics. Ann. Statist. 3, 119–131.Google Scholar
Rootzén, H., Leadbetter, M. R. and De Haan, L. (1998). On the distribution of tail array sums for strongly mixing stationary sequences. Ann. Appl. Prob. 8, 868–885.Google Scholar
Smith, R. L. (1992). The extremal index for a Markov chain. J. Appl. Prob. 29, 37–45.CrossRefGoogle Scholar
Smith, R. L. and Weissman, I. (1994). Estimating the extremal index. J. R. Statist. Soc. B 56, 515–528.Google Scholar
Smith, R. L., Tawn, J. A. and Coles, S. G. (1997). Markov chain models for threshold exceedances. Biometrika 84, 249–268.Google Scholar
Yun, S. (1998). The extremal index of a higher-order stationary Markov chain. Ann. Appl. Prob. 8, 408–437.Google Scholar
Yun, S. (2000). The distributions of cluster functionals of extreme events in a dth-order Markov chain. J. Appl. Prob. 37, 29–44.Google Scholar