A note on a limiting behaviour of the occurrence times of a mixed Poisson process

Abstract

In a recent publication Pfeifer (1982) shows that for a P61ya-Lundberg birth process the limiting distribution of the sequence {n/T,}, where T, denotes the nth occurrence time of the process, is a gamma distribution. As the P61ya-Lundberg birth process is a mixed Poisson process with a gamma mixing distribution, Pfeifer's result implies that the limiting distribution of {n/T,} is just the mixing distribution of the process. A stronger result of this type is valid for every mixed Poisson process, which can be seen as follows. Given a random variable A whose distribution is the mixing distribution of the process, we notice, following Grandell ((1976), p. 12), that the process N(t) = N(tA), where N(t) is a Poisson process with unit intensity, is a version of the desired mixed Poisson process. Hence ( N(t) N(At) P(im = A t= = P lim -= At) = 11.