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Strong memoryless times and rare events in Markov renewal point processesPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2004 (English)In: Annals of Probability, ISSN 0091-1798, Vol. 32, no 3B, 2446-2462 p.Article in journal (Refereed) Published
##### Abstract [en]

##### 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

Mathematics
##### Identifiers

URN: urn:nbn:se:liu:diva-23156DOI: 10.1214009117904000000054Local ID: 2560OAI: oai:DiVA.org:liu-23156DiVA: diva2:243470
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Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2011-01-12

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.

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