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 .