# 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 $\sigma_n\,\PP(S_n\!=\!j)\!\leq\! \eta$ 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 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.
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Combinatorics Probability and Computing, 2016
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https://hal.archives-ouvertes.fr/hal-01522068
Contributeur : Jean-Bernard Baillon <>
Soumis le : vendredi 12 mai 2017 - 17:34:24
Dernière modification le : lundi 27 novembre 2017 - 14:14:02

### Identifiants

• HAL Id : hal-01522068, version 1
• ARXIV : 0806.2350

### Citation

Jean-Bernard Baillon, Roberto Cominetti, José Vaisman. A Sharp Uniform Bound for the Distribution of Sums of Bernoulli Trials. Combinatorics Probability and Computing, 2016. 〈hal-01522068〉

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