Annals of Probability

Zero biasing and a discrete central limit theorem

Larry Goldstein and Aihua Xia

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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.

Article information

Ann. Probab., Volume 34, Number 5 (2006), 1782-1806.

First available in Project Euclid: 14 November 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G50: Sums of independent random variables; random walks

Stein’s method integer-valued random variables total variation


Goldstein, Larry; Xia, Aihua. Zero biasing and a discrete central limit theorem. Ann. Probab. 34 (2006), no. 5, 1782--1806. doi:10.1214/009117906000000250.

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