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A note on the point processes of rare events

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

J. Hüsler  and
M. Schmidt
J. Hüsler*
Affiliation:
University of Bern
M. Schmidt*
Affiliation:
University of Bern
*
∗Postal address for both authors: Universität Bern, Institut für Mathematische Statistik und Versicherungslehre, Sidlerstrasse 5, CH-3012 Bern, Switzerland.
∗Postal address for both authors: Universität Bern, Institut für Mathematische Statistik und Versicherungslehre, Sidlerstrasse 5, CH-3012 Bern, Switzerland.
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Abstract

We discuss the limits of point processes which are generated by a triangular array of rare events. Such point processes are motivated by the exceedances of a high boundary by a random sequence since exceedances are rare events in this case. This application relates the problem to extreme value theory from where the method is used to treat the asymptotic approximation of these point processes. The presented general approach extends, unifies and clarifies some of the various conditions used in the extreme value theory.

Keywords

RARE EVENTSEXTREMESEXCEEDANCESPOISSON PROCESSMIXING CONDITIONS

MSC classification

Primary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 3 , September 1996 , pp. 654 - 663
Copyright
Copyright © Applied Probability Trust 1996 

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References

Chernick, M. R., Hsing, T. and Mccormick, W. P. (1991) Calculating the extremal index for a class of stationary sequences. Adv. Appl. Prob. 23, 835–850.Google Scholar
Gänssler, P. and Stute, W. (1977) Wahrscheinlichkeitstheorie. Springer, Berlin.CrossRefGoogle Scholar
Hsing, T., Hüsler, J. and Reiss, R.-D. (1994) The extremes of a triangular array of normal random variables. Preprint. Google Scholar
Hüsler, J. (1983) Asymptotic approximations of crossing probabilities of random sequences. Z. Wahrscheinlichkeitsth. 63, 257–270.Google Scholar
Hüsler, J. (1986a) Extreme values and rare events of non-stationary random sequences. In Dependence in Probability and Statistics. ed. Eberlein, E. and Taqqu, M. S. Birkhäuser, Boston.Google Scholar
Hüsler, J. (1986b) Extreme values of non-stationary random sequences. J. Appl. Prob. 23, 937–950.Google Scholar
Hüsler, J. (1993) A note on exceedances and rare events of nonstationary sequences. J. Appl. Prob. 30, 877–888.Google Scholar
Kallenberg, O. (1986) Random Measures. 4th edn. Akademie, Berlin.Google Scholar
Leadbetter, M. R. (1974) On extreme values in stationary sequences. Z. Wahrscheinlichkeitsth. 28, 289–303.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer, New York.Google Scholar
Nandagopalan, S., Leadbetter, M. R. and Hüsler, J. (1993) Limit theorems for non-stationary multidimensional strongly mixing random measures. Preprint. Google Scholar
Reiss, R. D. (1993) A Course on Point Processes. Springer, New York.Google Scholar
Volkonskii, V. A. and Rozanov, Yu. A. (1959) Some limit theorems for random functions. Theory Prob. Appl 4, 178–197.Google Scholar