Identifiability in superpositions of renewal processes

Abstract

We are given a point process N which is known to be the superposition of two independent renewal processes. Can we deduce from the knowledge of the superposition alone what the distribution functions driving the renewal processes are? Clearly the answer is no if the two renewal processes are Poisson. Apart from this special case, and under the additional assumption of analyticity of densities, we prove that the answer is yes. This result answers a question posed by Neuts and Meier-Hellstern about superpositions of phase-type renewal processes. Using similar techniques we give examples showing that the well-known inclusion relations between certain families of point processes based on Markov chains are strict.