A sharp uniform bound for the distribution of sums of Bernoulli trials

TL;DR

This paper establishes a universal sharp bound for the probability distribution of sums of independent Bernoulli trials, including Poisson limits, with applications to convergence rates in fixed point iterations.

Contribution

The paper derives the best possible uniform bound for the distribution of sums of Bernoulli trials, extending to limits like Poisson laws, and provides applications to iterative convergence analysis.

Findings

Universal bound with constant ~0.4688 for Bernoulli sums

Bound applies to Poisson and other limit distributions

Application to Mann's fixed point iteration convergence

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 $\sigma_n \mathbb{P}(S_n\!=\!j)\leq\eta$ 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 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.