Bernoulli

Poisson process approximation for dependent superposition of point processes

Louis H.Y. Chen and Aihua Xia

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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.

Article information

Source
Bernoulli Volume 17, Number 2 (2011), 530-544.

Dates
First available: 5 April 2011

Permanent link to this document
http://projecteuclid.org/euclid.bj/1302009235

Digital Object Identifier
doi:10.3150/10-BEJ290

Mathematical Reviews number (MathSciNet)
MR2787603

Zentralblatt MATH identifier
06083980

Citation

Chen, Louis H.Y.; Xia, Aihua. Poisson process approximation for dependent superposition of point processes. Bernoulli 17 (2011), no. 2, 530--544. doi:10.3150/10-BEJ290. http://projecteuclid.org/euclid.bj/1302009235.


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