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Stein's method, Palm theory and Poisson process approximation | | Louis H. Y. Chen
; Aihua Xia
; | | Date: |
6 Oct 2004 | | Journal: | Annals of Probability 2004, Vol. 32, No. 3B, 2545-2569 DOI: 10.1214/009117904000000027 | | Subject: | Probability MSC-class: 60G55 (Primary) 60E15, 60E05 (Secondary) | math.PR | | 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
efimportantproposition) 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/lambda as in Poisson approximation, it provides good approximation, particularly in cases where lambda is large. The general result is applied to a number of problems including Poisson process modeling of rare words in a DNA sequence. | | Source: | arXiv, math.PR/0410169 | | Services: | Forum | Review | PDF | Favorites | |
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