Strong memoryless times and rare events in Markov renewal point processes

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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'' ζsuch that ζ-t is exponentially distributed conditional on ζ≤ 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.