Pairs of renewal processes whose superposition is a renewal process

A renewal process is called ordinary if its inter-renewal times are strictly positive. S.M. Samuels proved in 1974 that if the superposition of two ordinary renewal processes is an ordinary renewal process, then all processes are Poisson. This result is generalized here to the case of processes whose inter-renewal times may be zero. We show that, besides the Poisson processes, there are two pairs of binomial-like processes whose superposition is a renewal process. A new proof of Samuelsʹs theorem is included, which, unlike the original, does not require the renewal theorem. If the two processes are assumed identical, then a very simple proof is possible.