Stein's method and birth-death processes
type: Item
instance of: scholarly article
title: Stein's method and birth-death processes
zbMATH Open document ID: 1019.60019
author: Timothy C. Brown
author: Aihua Xia
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: The representation of a distribution \(Q\) on \(\mathbb{Z}_+\) as the equilibrium distribution of a birth and death process allows a particular realization of Stein's method for \(Q\). The probabilistic argument of \textit{A. Xia} [J. Appl. Probab. 36, No. 1, 287-290 (1999; Zbl 0942.60006)] is extended to yield bounds for the resulting `Stein factors': see also those of \textit{G. V. Weinberg} [ibid. 37, No. 4, 1181-1187 (2000; Zbl 0985.60018)]. The approach is also used to refine the very attractive non-uniform bounds for the second differences of the solutions to the Stein equation for a Poisson point process, in the case of \(d_2\)-test functions, which were originally derived by the authors and \textit{G. V. Weinberg} [Stochastic Processes Appl. 87, No. 1, 149-165 (2000)].
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60E05
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 1906021
zbMATH Keywords: equilibrium distribution
zbMATH Keywords: birth and death process
zbMATH Keywords: realization of Stein's method
zbMATH Keywords: Poisson point process
reviewed by: A. D. Barbour
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1015345606
OpenAlex ID: W2063452870
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DOI: 10.1214/AOP/1015345606
description: scientific article; zbMATH DE number 1906021