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July 2004 Stein’s method, Palm theory and Poisson process approximation
Louis H. Y. Chen, Aihua Xia
Ann. Probab. 32(3B): 2545-2569 (July 2004). DOI: 10.1214/009117904000000027

Abstract

The framework of Stein’s method for Poisson process approximation is presented from the point of view of Palm theory, which is used to construct Stein identities and define local dependence. A general result (Theorem 2.3) in Poisson process approximation is proved by taking the local approach. It is obtained without reference to any particular metric, thereby allowing wider applicability. A Wasserstein pseudometric is introduced for measuring the accuracy of point process approximation. The pseudometric provides a generalization of many metrics used so far, including the total variation distance for random variables and the Wasserstein metric for processes as in Barbour and Brown [Stochastic Process. Appl. 43 (1992) 9–31]. Also, through the pseudometric, approximation for certain point processes on a given carrier space is carried out by lifting it to one on a larger space, extending an idea of Arratia, Goldstein and Gordon [Statist. Sci. 5 (1990) 403–434]. The error bound in the general result is similar in form to that for Poisson approximation. As it yields the Stein factor 1/λ as in Poisson approximation, it provides good approximation, particularly in cases where λ is large. The general result is applied to a number of problems including Poisson process modeling of rare words in a DNA sequence.

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Louis H. Y. Chen. Aihua Xia. "Stein’s method, Palm theory and Poisson process approximation." Ann. Probab. 32 (3B) 2545 - 2569, July 2004. https://doi.org/10.1214/009117904000000027

Information

Published: July 2004
First available in Project Euclid: 6 August 2004

zbMATH: 1057.60051
MathSciNet: MR2078550
Digital Object Identifier: 10.1214/009117904000000027

Subjects:
Primary: 60G55
Secondary: 60E05 , 60E15

Keywords: local approach , local dependence , Palm process , point process , Poisson process approximation , Stein’s method , Wasserstein pseudometric

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 3B • July 2004
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