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September 2006 Zero biasing and a discrete central limit theorem
Larry Goldstein, Aihua Xia
Ann. Probab. 34(5): 1782-1806 (September 2006). DOI: 10.1214/009117906000000250


We introduce a new family of distributions to approximate ℙ(WA) for A⊂{…, −2, −1, 0, 1, 2, …} and W a sum of independent integer-valued random variables ξ1, ξ2, …, ξn with finite second moments, where, with large probability, W is not concentrated on a lattice of span greater than 1. The well-known Berry–Esseen theorem states that, for Z a normal random variable with mean $\mathbb {E}(W)$ and variance Var (W), ℙ(ZA) provides a good approximation to ℙ(WA) for A of the form (−∞, x]. However, for more general A, such as the set of all even numbers, the normal approximation becomes unsatisfactory and it is desirable to have an appropriate discrete, nonnormal distribution which approximates W in total variation, and a discrete version of the Berry–Esseen theorem to bound the error. In this paper, using the concept of zero biasing for discrete random variables (cf. Goldstein and Reinert [J. Theoret. Probab. 18 (2005) 237–260]), we introduce a new family of discrete distributions and provide a discrete version of the Berry–Esseen theorem showing how members of the family approximate the distribution of a sum W of integer-valued variables in total variation.


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Larry Goldstein. Aihua Xia. "Zero biasing and a discrete central limit theorem." Ann. Probab. 34 (5) 1782 - 1806, September 2006.


Published: September 2006
First available in Project Euclid: 14 November 2006

zbMATH: 1111.60015
MathSciNet: MR2271482
Digital Object Identifier: 10.1214/009117906000000250

Primary: 60F05
Secondary: 60G50

Keywords: integer-valued random variables , Stein’s method , Total variation

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 5 • September 2006
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