Poisson process approximation for dependent superposition of point processes

Although the study of weak convergence of superposition of point processes to the Poisson process dates back to the work of Grigelionis in 1963, it was only recently that Schuhmacher (2005a) obtained error bounds for the weak convergence. Schuhmacher considered dependent superposition, truncated the individual point processes to 0– 1 point processes and then applied Stein’s method to the latter. In this paper we take a different approach to the problem by using Palm theory and Stein’s method, thereby expressing the error bounds in terms of the mean measures of the individual point processes, which is not possible by Schuhmacher’s approach. We consider locally dependent superposition as a generalization of the locally dependent point process introduced in Chen and Xia (2004) and apply the main theorem to the superposition of thinned point processes and of renewal processes.