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Occupation times of alternating renewal processes with Lévy applications

Part of: Stochastic processes Operations research and management science Special processes

Published online by Cambridge University Press:  16 January 2019

Nicos Starreveld ,
Réne Bekker  and
Michel Mandjes
Nicos Starreveld*
Affiliation:
University of Amsterdam
Réne Bekker*
Affiliation:
Vrije Universiteit Amsterdam
Michel Mandjes*
Affiliation:
University of Amsterdam
*
* Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands.
*** Postal address: Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands. Email address: r.bekker@vu.nl
* Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands.
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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)∕t≥q). We apply our findings to spectrally positive Lévy processes reflected at the infimum and establish various new occupation time results for the corresponding model.

Keywords

Occupation timealternating renewal processLévy processreflected Brownian motioncentral limit theoremlarge deviations

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes 60G51: Processes with independent increments; L'evy processes
Secondary: 90B05: Inventory, storage, reservoirs
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1287 - 1308
Copyright
Copyright © Applied Probability Trust 2018 

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