Strong memoryless times and rare events in Markov renewal point processes

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dc.creatorErhardsson, Torkel
dc.date2004-10-06
dc.date.accessioned2026-07-07T05:12:59Z
dc.date.available2026-07-07T05:12:59Z
dc.descriptionLet W be the number of points in (0,t] of a stationary finite-state Markov renewal point process. We derive a bound for the total variation distance between the distribution of W and a compound Poisson distribution. For any nonnegative random variable ζ, we construct a ``strong memoryless time'' \hat ζsuch that ζ-t is exponentially distributed conditional on {\hat ζ\leq t, ζ>t}, for each t. This is used to embed the Markov renewal point process into another such process whose state space contains a frequently observed state which represents loss of memory in the original process. We then write W as the accumulated reward of an embedded renewal reward process, and use a compound Poisson approximation error bound for this quantity by Erhardsson. For a renewal process, the bound depends in a simple way on the first two moments of the interrenewal time distribution, and on two constants obtained from the Radon-Nikodym derivative of the interrenewal time distribution with respect to an exponential distribution. For a Poisson process, the bound is 0.
dc.descriptionPublished by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000054
dc.identifierhttps://arxiv.org/abs/math/0410166
dc.identifierhttp://arxiv.org/abs/math/0410166
dc.identifierAnnals of Probability 2004, Vol. 32, No. 3B, 2446-2462
dc.identifierdoi:10.1214/009117904000000054
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/72782
dc.subjectProbability
dc.subject60K15 (Primary) 60E15 (Secondary)
dc.titleStrong memoryless times and rare events in Markov renewal point processes
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