Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T03:48:55.953Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on exceedances and rare events of non-stationary sequences

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

J. Hüsler
J. Hüsler*
Affiliation:
University of Bern
*
∗ Postal address: Department of Mathematical Statistics, University of Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland.
Get access Rights & Permissions [Opens in a new window]

Abstract

Exceedances of a non-stationary sequence above a boundary define certain point processes, which converge in distribution under mild mixing conditions to Poisson processes. We investigate necessary and sufficient conditions for the convergence of the point process of exceedances, the point process of upcrossings and the point process of clusters of exceedances. Smooth regularity conditions, as smooth oscillation of the non-stationary sequence, imply that these point processes converge to the same Poisson process. Since exceedances are asymptotically rare, the results are extended to triangular arrays of rare events.

Keywords

LEVEL CROSSINGSTRIANGULAR ARRAYS

MSC classification

Primary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 4 , December 1993 , pp. 877 - 888
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

Hüsler, J. (1986a) Extreme values of non-stationary random sequences. J. Appl. Prob. 23, 937–950.Google Scholar
Hüsler, J. (1986b) Extreme values and rare events of non-stationary random sequences. In Dependence in Probability and Statistics, ed. Eberlein, E. and Taqqu, M. S., pp. 439–456. Birkhäuser, Boston.Google Scholar
Kallenberg, O. (1983) Random Measures, 2nd edn. Akademie-Verlag, Berlin.Google Scholar
Leadbetter, M. R. and Nandagopalan, S. (1990) On exceedance point processes for stationary sequences under mild oscillation restrictions. In Extreme Value Theory. Lecture Notes in Statistics 51, eds. Hüsler, J. and Reiss, R.-D., pp. 69–80. Springer-Verlag, Berlin.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.Google Scholar
Nandagopalan, S., Leadbetter, M. R. and Hüsler, J. (1991) Limit theorems of multidimensional random measures. To be published.Google Scholar