On the Superposition of Point Processes

Abstract

Summary Given point processes N1, …, Nm their superposition is the point process N defined by N(t) = N1(t) + … + Nm(t), t ≥ 0. An equivalent description of the system (N1, …, Nm) is by the process (Xn, Tn) where the Tn are the points of N, and Xn = k if and only if Tn is a point of Nk. The use of (X, T) process enables one to study the dependence of N1, …, Nm. Necessary and sufficient conditions are obtained for N, …, Nm to be independent, and for the superposition N to be a renewal process. For example, if N1 and N2 are renewal processes and X is independent of N, then N is a renewal process only if X is a homogeneous Markov chain.