Mean absolute deviations of sample means and minimally concentrated binomials

Abstract

This is a contribution to the theory of sums of independent random variables at the level of optimal explicit inequalities: we compute the optimal constants in Hornich's lower bounds for the mean absolute deviations of sample means. This is done by reducing the original problem to the elementary one of determining the minimally concentrated binomial distributions $B_{n,p}$ with fixed sample size parameter $n$.