Some characterizations of the Poisson process and geometric renewal process

Abstract

Let γ t and δ t denote the residual life at t and current life at t, respectively, of a renewal process , with the sequence of interarrival times. We prove that, given a function G, under mild conditions, as long as holds for a single positive integer n, then is a Poisson process. On the other hand, for a delayed renewal process with the residual life at t, we find that for some fixed positive integer n, if is independent of t, then is an arbitrarily delayed Poisson process. We also give some corresponding results about characterizing the common distribution function F of the interarrival times to be geometric when F is discrete. Finally, we obtain some characterization results based on the total life or independence of γ t and δ t.