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April 1998 On the excursion random measure of stationary processes
Tailen Hsing, M. R. Leadbetter
Ann. Probab. 26(2): 710-742 (April 1998). DOI: 10.1214/aop/1022855648


The excursion random measure $\zeta$ of a stationary process is defined on sets $E \subset (-\infty, \infty) \times (0, \infty)$, as the time which the process (suitably normalized) spends in the set E . Particular cases thus include a multitude of features (including sojourn times) related to high levels. It is therefore not surprising that a single limit theorem for $\zeta$ at high levels contains a wide variety of useful extremal and high level exceedance results for the stationary process itself.

The theory given for the excursion random measure demonstrates, under very general conditions, its asymptotic infinite divisibility with certain stability and independence of increments properties leading to its asymptotic distribution (Theorem 4.1). The results are illustrated by a number of examples including stable and Gaussian processes.


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Tailen Hsing. M. R. Leadbetter. "On the excursion random measure of stationary processes." Ann. Probab. 26 (2) 710 - 742, April 1998.


Published: April 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0935.60013
MathSciNet: MR1626519
Digital Object Identifier: 10.1214/aop/1022855648

Primary: 60F05 , 60G10

Keywords: Extremes , Infinite divisibility , sojourns , weak convergence

Rights: Copyright © 1998 Institute of Mathematical Statistics


Vol.26 • No. 2 • April 1998
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