The Annals of Probability

Stein's Method and Birth-Death Processes

Timothy C. Brown and Aihua Xia

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Barbour introduced a probabilistic view of Stein's method for estimating the error in probability approximations. However, in the case of approximations by general distributions on the integers, there have been no purely probabilistic proofs of Stein's bounds till this paper. Furthermore, the methods introduced here apply to a very large class of approximating distributions on the non-negative integers, among which there is a natural class for higher-order approximations by probability distributions rather than signed measures (as previously). The methods also produce Stein magic factors for process approximations which do not increase with the window of observation and which are simpler to apply than those in Brown, Weinberg and Xia.

Article information

Ann. Probab., Volume 29, Number 3 (2001), 1373-1403.

First available in Project Euclid: 5 March 2002

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Zentralblatt MATH identifier

Primary: 60E05: Distributions: general theory
Secondary: 60E15: Inequalities; stochastic orderings 60F05: Central limit and other weak theorems 60G55: Point processes

Stein's method birth-death process distributional approximation total variation distance Poisson process approximation Wasserstein distance compound Poisson distribution negative binomial distribution polynomial birth-death distribution


Brown, Timothy C.; Xia, Aihua. Stein's Method and Birth-Death Processes. Ann. Probab. 29 (2001), no. 3, 1373--1403. doi:10.1214/aop/1015345606.

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