The Annals of Probability

Stein’s method, Palm theory and Poisson process approximation

Louis H. Y. Chen and Aihua Xia

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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

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

First available in Project Euclid: 6 August 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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