Poisson process approximation for dependent superposition of point 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.

Published in at http://dx.doi.org/10.3150/10-BEJ290 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (3)

• Normal approximation under local dependence
• Stein's method, Palm theory and Poisson process approximation
• Zero biasing and a discrete central limit theorem

Cited by in corpus (3)

• A Generalization of the Space-Fractional Poisson Process and its Connection to some Lévy Processes
• Normal approximation for fire incident simulation using permanental Cox processes
• Normal approximation for associated point processes via Stein's method with applications to determinantal point processes