{
  "id": "math/0509444",
  "version": "v2",
  "published": "2005-09-20T05:02:08.000Z",
  "updated": "2006-11-22T06:38:34.000Z",
  "title": "Zero biasing and a discrete central limit theorem",
  "authors": [
    "Larry Goldstein",
    "Aihua Xia"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/009117906000000250 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2006, Vol. 34, No. 5, 1782-1806",
  "doi": "10.1214/009117906000000250",
  "categories": [
    "math.PR"
  ],
  "abstract": "We introduce a new family of distributions to approximate $\\mathbb {P}(W\\in A)$ for $A\\subset\\{...,-2,-1,0,1,2,...\\}$ and $W$ a sum of independent integer-valued random variables $\\xi_1$, $\\xi_2$, $...,$ $\\xi_n$ 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 $\\operatorname {Var}(W)$, $\\mathbb {P}(Z\\in A)$ provides a good approximation to $\\mathbb {P}(W\\in A)$ for $A$ of the form $(-\\infty,x]$. 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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-11-22T06:38:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60G50"
    ],
    "keywords": [
      "discrete central limit theorem",
      "zero biasing",
      "distribution",
      "total variation",
      "discrete version"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......9444G"
    }
  }
}