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 $S_n=X_1+\cdots +X_n$ of independent non-homogeneous Bernoulli trials. Specifically, we prove that ${\sigma }_n\mathbb{P}(S_n=j)\le \eta$ 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 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.

arXiv commentThis paper is a revised version of a previous article

References in corpus (1)

• On the rate of convergence of Krasnoselski-Mann iterations and their connection with sums of Bernoullis

Cited by in corpus (1)

• Improved Runtime Bounds for the Univariate Marginal Distribution Algorithm via Anti-Concentration