Poisson process approximation for dependent superposition of point processes

TL;DR

This paper introduces a novel method using Palm theory and Stein's method to derive error bounds for the weak convergence of dependent superpositions of point processes to the Poisson process, improving upon previous approaches.

Contribution

It presents a new approach leveraging Palm theory and Stein's method to express error bounds in terms of mean measures, applicable to locally dependent superpositions.

Findings

Derived explicit error bounds for dependent superpositions

Applied the main theorem to thinned point processes

Extended the framework to renewal processes

Abstract

Although the study of weak convergence of superpositions of point processes to the Poisson process dates back to the work of Grigelionis in 1963, it was only recently that Schuhmacher [Stochastic Process. Appl. 115 (2005) 1819--1837] 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 adopt 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 with Schuhmacher's approach. We consider locally dependent superposition as a generalization of the locally dependent point process introduced in Chen and Xia [Ann. Probab. 32 (2004) 2545--2569] and apply the main theorem to the superposition of thinned point processes and of renewal processes.