The Annals of Probability

Stein’s method, Palm theory and Poisson process approximation

Louis H. Y. Chen and Aihua Xia

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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.

Article information

Source
Ann. Probab., Volume 32, Number 3B (2004), 2545-2569.

Dates
First available in Project Euclid: 6 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1091813623

Digital Object Identifier
doi:10.1214/009117904000000027

Mathematical Reviews number (MathSciNet)
MR2078550

Zentralblatt MATH identifier
1057.60051

Subjects
Primary: 60G55: Point processes
Secondary: 60E15: Inequalities; stochastic orderings 60E05: Distributions: general theory

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

Citation

Chen, Louis H. Y.; Xia, Aihua. Stein’s method, Palm theory and Poisson process approximation. Ann. Probab. 32 (2004), no. 3B, 2545--2569. doi:10.1214/009117904000000027. https://projecteuclid.org/euclid.aop/1091813623


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