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Occupation times of alternating renewal processes with Lévy applications

  • Nicos Starreveld (a1), Réne Bekker (a2) and Michel Mandjes (a1)

Abstract

In this paper we present a set of results relating to the occupation time α(t) of a process X(·). The first set of results concerns exact characterizations of α(t), e.g. in terms of its transform up to an exponentially distributed epoch. In addition, we establish a central limit theorem (entailing that a centered and normalized version of α(t)∕t converges to a zero-mean normal random variable as t→∞) and the tail asymptotics of ℙ(α(t)∕tq). We apply our findings to spectrally positive Lévy processes reflected at the infimum and establish various new occupation time results for the corresponding model.

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Corresponding author

* Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands.
** Email address: n.j.starreveld@uva.nl
*** Postal address: Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands. Email address: r.bekker@vu.nl
**** Email address: m.r.h.mandjes@uva.nl

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