Pairs of renewal processes whose superposition is a renewal process

A renewal process is called ordinary if its inter-renewal times are strictl y positive. S.M. Samuels proved in 1974 that if the superposition of two or dinary renewal processes is an ordinary renewal process, then all processes are Poisson. This result is generalized here to the case of processes whos e inter-renewal times may be zero. We show that, besides the Poisson proces ses, there are two pairs of binomial-like processes whose superposition is a renewal process. A new proof of Samuels's theorem is included, which, unl ike the original, does not require the renewal theorem. If the two processe s are assumed identical, then a very simple proof is possible. (C) 2000 Els evier Science B.V. All rights reserved.