Zero biasing and a discrete central limit theorem

Abstract

We introduce a new family of distributions to approximate ℙ(W∈A) for A⊂{…, −2, −1, 0, 1, 2, …} and W a sum of independent integer-valued random variables ξ₁, ξ₂, …, ξ_n 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(Z∈A) provides a good approximation to ℙ(W∈A) 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.