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Strong memoryless times and rare events in Markov renewal point processes
KTH, Superseded Departments, Mathematics.
2004 (English)In: Annals of Probability, ISSN 0091-1798, Vol. 32, no 3B, 2446-2462 p.Article in journal (Refereed) Published
Abstract [en]

Let 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 (, we construct a "strong memoryless time" zeta such that zeta - t is exponentially distributed conditional on {zeta less than or equal to t, zeta > 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.

Place, publisher, year, edition, pages
2004. Vol. 32, no 3B, 2446-2462 p.
Keyword [en]
strong memoryless time, Markov renewal process, number of points, rare event, compound Poisson, approximation, error bound
National Category
URN: urn:nbn:se:kth:diva-41887DOI: 10.1214/009117904000000054ISI: 000223557000005ScopusID: 2-s2.0-4544383957OAI: diva2:450845
QC 20111023Available from: 2011-10-23 Created: 2011-10-03 Last updated: 2011-10-23Bibliographically approved

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