Abstract.
We consider a stationary version of a renewal reward process, i.e., a renewal process where a random variable called a reward is associated with each renewal. The rewards are nonnegative and I.I.D., but each reward may depend on the distance to the next renewal. We give an explicit bound for the total variation distance between the distribution of the accumulated reward over the interval (0,L] and a compound Poisson distribution. The bound depends in its simplest form only on the first two joint moments of T and Y (or I{Y > 0}), where T is the distance between successive renewals and Y is the reward. If T and Y are independent, and LE(Y) (or LP(Y > 0)) is bounded or Y binary valued, then the bound is O(E(Y)) as E(Y) → 0 (or O(P(Y > 0)) as P(Y > 0) → 0). To prove our result we generalize a Poisson approximation theorem for point processes by Barbour and Brown, derived using Stein's method and Palm theory, to the case of compound Poisson approximation, and combine this theorem with suitable couplings.
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Received: 1 March 1999 / Revised version: 2 August 1999 /¶Published online: 31 May 2000
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Erhardsson, T. On stationary renewal reward processes where most rewards are zero. Probab Theory Relat Fields 117, 145–161 (2000). https://doi.org/10.1007/s004400050001
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DOI: https://doi.org/10.1007/s004400050001