On normal approximations to symmetric hypergeometric laws

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The Kolmogorov distances between a symmetric hypergeometric law with standard deviation σ and its usual normal approximations are computed and shown to be less than 1/(√8πσ), with the order 1/σ and the constant 1/√8π 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 Lévy's concentration-variance inequality.