Stein’s method, Palm theory and Poisson process approximation

By Louis H. Y. Chen, Aihua Xia. 2004

50 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/901707c7c1f24dd402d8edf46216a675a1a80f97

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In Applied Mathematics

In Discrete Mathematics and Combinatorics

In Statistics and Probability

Citations in the corpus

Exchangeable pairs and Poisson approximation (2004, background): “For a connection with Palm theory, see [15].”
Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry (2014, unknown): “tion), one often uses weaker notions of distance. Starting with the work of Barbour and Brown [4] and 2 Barbour, Holst and Janson [8], this has been done by Brown, Chen, Schuhmacher, Weinberg and Xia [11, 12, 13, 15, 39, 40, 42]. Our work goes in the opposite direction since the KantorovichRubinstei”
A new metric between distributions of point processes (2007, background): “Numerous useful upper bounds in this metric have been obtained, including the results of [3], [6], [9], [11], [12], [22], and [23], which for the most part assume that one of the probability measures involved is a Poisson (or compound Poisson) process distribution.”