On normal approximations to symmetric hypergeometric laws

TL;DR

This paper provides precise bounds on how closely symmetric hypergeometric distributions can be approximated by normal distributions, with optimal constants and connections to existing binomial results and broader probabilistic inequalities.

Contribution

It derives optimal bounds for the Kolmogorov distance between symmetric hypergeometric laws and their normal approximations, extending known results and connecting to Berry-Esseen type inequalities.

Findings

Kolmogorov distance less than 1/(√(8π)σ) for symmetric hypergeometric laws

Extension of Hipp and Mattner's results to hypergeometric distributions

Includes sharp inequalities and insights into related probabilistic inequalities

Abstract

The Kolmogorov distances between a symmetric hypergeometric law with standard deviation $\sigma$ and its usual normal approximations are computed and shown to be less than $1/(\sqrt{8\pi}\,\sigma)$, with the order $1/\sigma$ and the constant $1/\sqrt{8\pi}$ being optimal. The results of Hipp and Mattner (2007) for symmetric binomial laws are obtained as special cases. Connections to Berry-Esseen type results in more general situations concerning sums of simple random samples or Bernoulli convolutions are explained. Auxiliary results of independent interest include rather sharp normal distribution function inequalities, a simple identifiability result for hypergeometric laws, and some remarks related to L\'evy's concentration-variance inequality.