Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T06:39:55.813Z Has data issue: false hasContentIssue false
  • English
  • Français

Joint exceedances of high levels under a local dependence condition

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Helena Ferreira
Helena Ferreira*
Affiliation:
University of Coimbra
*
∗ Postal address: Department of Mathematics, University of Coimbra CMUC (INIC), Apartado 3008, 3000 Coimbra, Portugal.
Get access Rights & Permissions [Opens in a new window]

Abstract

Under appropriate long-range dependence conditions, it is well known that the joint distribution of the number of exceedances of several high levels is asymptotically compound Poisson. Here we investigate the structure of a cluster of exceedances for stationary sequences satisfying a suitable local dependence condition, under which it is only necessary to get certain limiting probabilities, easy to compute, in order to obtain limiting results for the highest order statistics, exceedance counts and upcrossing counts.

Keywords

STATIONARY SEQUENCEPOINT PROCESSES OF EXCEEDANCES AND UPCROSSINGSCLUSTERING OF EXTREME VALUES

MSC classification

Primary: 60G10: Stationary processes
Secondary: 60G55: Point processes 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 1 , March 1993 , pp. 112 - 120
Copyright
Copyright © Applied Probability Trust 1993 

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

[1] Chernick, M. R. (1981) On strong mixing and Leadbetter's D condition. J. Appl. Prob. 18, 764–769.Google Scholar
[2] Cohen, J. P. (1989) On the compound Poisson limit for high level exceedances. J. Appl. Prob. 26, 458–465.Google Scholar
[3] Davison, A. C. and Smith, R. L. (1990) Models for exceedances over high thresholds. J. R. Statist. Soc. B52, 393–442.Google Scholar
[4] Hsing, T. (1988) On the extreme order statistics for a stationary sequence. Stoch. Proc. Appl. 29, 155–169.CrossRefGoogle Scholar
[5] Hsing, T., Hüsler, J. and Leadbetter, M. R. (1988) On the exceedance point process for stationary sequence. Prob. Theory Rel. Fields 78, 97–112.CrossRefGoogle Scholar
[6] Kallenberg, O. (1983) Random Measures, 3rd edn. Akademie-Verlag, Berlin; Academic Press, London.Google Scholar
[7] Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, Berlin.Google Scholar
[8] Leadbetter, M. R. and Nandagopalan, S. (1989) On exceedances of point processes for stationary sequences under mild oscillation restrictions. In Extreme Values, ed. Hüsler, J. and Reiss, R.-D., pp. 69–80. Springer-Verlag, Berlin.Google Scholar