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

Jean-Bernard Baillon (1) , Roberto Cominetti (2) , José Vaisman (3)

1 SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)
2 DII - Departamento de Ingenieria Industrial [Santiago]
3 DIM - Departamento de Ingeniera Matematica [Santiago]

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.

Mots clés: Bernoulli; Mann's iteration; non expansive mapping; fixed point

Domaines: Analyse fonctionnelle [math.FA]

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⟩