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

International audienceIn this note we establish a uniform bound for the distribution of a sum$S_n=X_1+\cdots+X_n$ of independent non-homogeneous Bernoulli trials. Specifically, we prove that $\sigma_n\,\PP(S_n\!=\!j)\!\leq\! \eta$ where $\sigma_n$ denotes the standard deviation of $S_n$ and $\eta$ is a universal constant.We compute the best possible constant $\eta\!\sim\! 0.4688$ and we show thatthe 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 theoptimal bounds for $n$ and $j$ fixed.An application to estimate the rate of convergence of Mann's fixed point iterations is presented