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Strong memoryless times and rare events in Markov renewal point processes
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematical Statistics .
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 ren ewal point process. We derive a bound for the total variation distance between the distribution of $W$ and a compound Poisson distribution. For any nonnegative rand om variable $\zeta$ we construct a ``strong memoryless time'' $\hat\zeta$ such tha t $\zeta-t$ is exponentially distributed conditional on $\{\hat\zeta\leq t,\zeta>t \}$, for each $t$. This is used to embed the Markov renewal point process into ano ther such process whose state space contains a frequently observed state which rep resents loss of memory in the original process. We then write $W$ as the accumulat ed reward of an embedded renewal reward process, and use a compound Poisson approx imation error bound for this quantity by Erhardsson. For a renewal process, the bo und depends in a simple way on the first two moments of the interrenewal time dist ribution, and on two constants obtained from the Radon-Nikodym derivative of the i nterrenewal 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:liu:diva-23156DOI: 10.1214009117904000000054Local ID: 2560OAI: diva2:243470
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2011-01-12

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Erhardsson, Torkel
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The Institute of TechnologyMathematical Statistics
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