## The Annals of Probability

### Strong memoryless times and rare events in Markov renewal point processes

Torkel Erhardsson

#### Abstract

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 ζ, we construct a “strong memoryless time” $\hat{\zeta}$ such that ζ−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 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.

#### Article information

Source
Ann. Probab., Volume 32, Number 3B (2004), 2446-2462.

Dates
First available in Project Euclid: 6 August 2004

https://projecteuclid.org/euclid.aop/1091813619

Digital Object Identifier
doi:10.1214/009117904000000054

Mathematical Reviews number (MathSciNet)
MR2078546

Zentralblatt MATH identifier
1058.60070

Subjects
Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60E15: Inequalities; stochastic orderings

#### Citation

Erhardsson, Torkel. Strong memoryless times and rare events in Markov renewal point processes. Ann. Probab. 32 (2004), no. 3B, 2446--2462. doi:10.1214/009117904000000054. https://projecteuclid.org/euclid.aop/1091813619

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