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On the compound Poisson limit for high level exceedances

Published online by Cambridge University Press:  14 July 2016

Jonathan Cohen
Jonathan Cohen*
Affiliation:
University of Kentucky
*
∗Present address: Systems Applications Inc., 101 Lucas Valley Road, San Rafael, CA 94903, USA.
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Abstract

Let ∊1∊2, · ··, be a stationary sequence satisfying the weak long-range dependence condition Δ (un(τ)) of [3] for every τ > 0, where nP(∊1> un(τ))→ τ. Assume only that P (there are j exceedances of un(τ) by ∊1, ∊2, · ··, ∊n) converges for all j with 0≦j≦υ<∞ and a given fixedτ. Then the same holds for every τ> 0. For 0≦j≦υ the limit is P(X = j) where X is compound Poisson and the multiplicity distribution is independent ofτ. These results are extended to more general levels un and to cases where the joint distribution of the numbers of exceedances of several levels is considered. The limiting distributions of linearly normalized extreme order statistics are derived as a corollary. An application to insurance claim data is discussed.

Keywords

EXTREME VALUE THEORYCOMPOUND POISSON DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 3 , September 1989 , pp. 458 - 465
Copyright
Copyright © Applied Probability Trust 1989 

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References

[1] Cohen, J. P. (1987) On the compound Poisson limit for an exceedance process. Technical Report 259, Department of Statistics, University of Kentucky.Google Scholar
[2] Hsing, T. (1984) Point Processes associated with Extreme Value Theory. Ph.D. Dissertation and Technical Report 83, Department of Statistics, University of North Carolina.Google Scholar
[3] 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.Google Scholar
[4] Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, Berlin.Google Scholar
[5] Teugels, J. L. (1984) Extreme values in insurance mathematics. In Statistical Extremes and Applications, ed. Tiago de Oliveira, J., D. Reidel, Dordrecht, 253–259.CrossRefGoogle Scholar