Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T23:30:04.050Z Has data issue: false hasContentIssue false
  • English
  • Français

The distributions of cluster functionals of extreme events in a dth-order Markov chain

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Seokhoon Yun
Seokhoon Yun*
Affiliation:
University of Suwon
*
∗Postal address: Department of Applied Statistics, University of Suwon, Suwon, Kyonggi-do 445–743, South Korea. Email address: syun@stat.suwon.ac.kr
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper concerns the asymptotic distributions of cluster functionals of extreme events in a dth-order stationary Markov chain {Xn, n = 1,2,…} for which the joint distribution of (X1,…,Xd+1) is absolutely continuous. Under some distributional assumptions for {Xn}, we establish weak convergence for a class of cluster functionals and obtain representations for the asymptotic distributions which are well suited for simulation. A number of examples important in applications are presented to demonstrate the usefulness of the results.

Keywords

Extreme eventscluster functionalsdth-order stationary Markov chain

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60G10: Stationary processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 1 , March 2000 , pp. 29 - 44
Copyright
Copyright © 2000 by The Applied Probability Trust 

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.)

References

Hsing, T., Hüsler, J., and Leadbetter, M. R. (1988). On the exceedance point process for a stationary sequence. Prob. Theory Rel. Fields 78, 97–112.CrossRefGoogle Scholar
Leadbetter, M. R. (1991). On a basis for ‘peaks over threshold’ modeling. Statist. & Prob. Lett. 12, 357–362.Google Scholar
Leadbetter, M. R. (1995). On high level exceedance modeling and tail inference. J. Statist. Plann. Inference 45, 247–260.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
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.Google Scholar
Perfekt, R. (1995). Local extreme behaviour of dth order Markov chains. Preprint, University of Lund, Sweden.Google Scholar
Resnick, S. I. (1987). Extreme Values, Point Processes and Regular Variation. Springer, New York.Google Scholar
Rootzén, H. (1988). Maxima and exceedances of stationary Markov chains. Adv. Appl. Prob. 20, 371–390.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.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. (1997). On domains of attraction of multivariate extreme value distributions under absolute continuity. J. Multivar. Anal. 63, 277–295.Google Scholar
Yun, S. (1998). The extremal index of a higher-order stationary Markov chain. Ann. Appl. Prob. 8, 408–437.Google Scholar