A Sharp Uniform Bound for the Distribution of Sums of Bernoulli Trials

Abstract

In this note we establish a uniform bound for the distribution of a sum Sn=X1+···+Xn of independent non-homogeneous Bernoulli trials. Specifically, we prove that σn(Sn = j) ≤ η, where σn denotes the standard deviation of Sn, and η is a universal constant. We compute the best possible constant η ~ 0.4688 and we show that the bound also holds for limits of sums and differences of Bernoullis, including the Poisson laws which constitute the worst case and attain the bound. We also investigate the optimal bounds for n and j fixed. An application to estimate the rate of convergence of Mann's fixed-point iterations is presented.