TY - JOUR

T1 - A sharp uniform bound for the distribution of sums of Bernoulli trials

AU - Baillon, Jean Bernard

AU - Cominetti, Roberto

AU - Vaisman, José

N1 - Funding Information:
We are indebted to Professor David McDonald for helpful discussions on the connection of our main result with the local limit theorem, as well as to Professor Michel Weber for pointing out the Kolmogorov-Rogozin inequality. We also thank an anonymous referee for very useful suggestions that contributed to improvement of the presentation. Roberto Cominetti was supported by FONDECYT grant 1100046 (CONICYT-Chile), as well as Nucleo Milenio Informacion y Coordinacionen Redes ICM/FIC P10-024F. This work was completed during a visit by Jean-Bernard Baillon to Universidad de Chile, which was supported by FONDECYT grant 1130564.
Publisher Copyright:
Copyright © Cambridge University Press 2015.

PY - 2016/5/1

Y1 - 2016/5/1

N2 - In this note we establish a uniform bound for the distribution of a sum S n=X 1+···+X n of independent non-homogeneous Bernoulli trials. Specifically, we prove that σ n (S n = j) ≤ η, where σ n denotes the standard deviation of S n, and η is a universal constant. We compute the best possible constant η ~ 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.

AB - In this note we establish a uniform bound for the distribution of a sum S n=X 1+···+X n of independent non-homogeneous Bernoulli trials. Specifically, we prove that σ n (S n = j) ≤ η, where σ n denotes the standard deviation of S n, and η is a universal constant. We compute the best possible constant η ~ 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.

UR - http://www.scopus.com/inward/record.url?scp=84935455121&partnerID=8YFLogxK

U2 - 10.1017/S0963548315000127

DO - 10.1017/S0963548315000127

M3 - Article

AN - SCOPUS:84935455121

VL - 25

SP - 352

EP - 361

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 3

ER -