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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.
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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 

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References

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