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

Part of: Distribution theory - Probability Combinatorial probability

Published online by Cambridge University Press:  07 July 2015

JEAN-BERNARD BAILLON ,
ROBERTO COMINETTI  and
JOSÉ VAISMAN
JEAN-BERNARD BAILLON
Affiliation:
SAMM–EA 4543, Université de Paris 1, 75013 Paris, France (e-mail: baillon@univ.paris1.fr)
ROBERTO COMINETTI
Affiliation:
Departamento de Ingeniería Industrial, Universidad de Chile, Avenida República 701, Santiago, Chile (e-mail: cominetti.roberto@gmail.com)
JOSÉ VAISMAN
Affiliation:
Departamento de Ingeniería Matemática, Universidad de Chile, Avenida Blanco Encalada 2120, Santiago, Chile (e-mail: hellovaisman@gmail.com)
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Abstract

In this note we establish a uniform bound for the distribution of a sum Sn=X1+···+Xn of independent non-homogeneous Bernoulli trials. Specifically, we prove that σn(Sn = j) ≤ η, where σn denotes the standard deviation of Sn, 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.

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60E05: Distributions 60E15: Inequalities; stochastic orderings
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 3 , May 2016 , pp. 352 - 361
Copyright
Copyright © Cambridge University Press 2015 

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