Renewal processes decomposable into i.i.d. components

Abstract

Let N be a stationary renewal process with a probability density function f(t). Suppose that N can be expressed as the superposition of a finite number of i.i.d. stationary components N(1), …, N(p) (p≧2). Then, under a supplementary condition on f(t), N and N(1), …, N(p) are all Poisson. This is proved by using recurrence relations given in Ito (1978) for the probability distribution of i.i.d. components of a superposition process.