ON NORMAL APPROXIMATIONS TO SYMMETRIC HYPERGEOMETRIC LAWS

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/(root 8 pi sigma), with the order 1/sigma and the constant 1/root 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 Levy's concentration-variance inequality.