A Sharp Uniform Bound for the Distribution of Sums of Bernoulli Trials. (7th July 2015)

• Record Type: Journal Article
• Authors: BAILLON, JEAN-BERNARD; COMINETTI, ROBERTO; VAISMAN, JOSÉ
• Is Part Of: Combinatorics, probability and computing. Volume 25:Number 3(2016:May)
• Page Start: 352
• Page End: 361
• Publication Date: 2015-07-07
• Subjects: Primary 60C05, -- Secondary 60E05, -- 60E15; Combinatorial analysis -- Periodicals; Probabilities -- Periodicals; Computer science -- Mathematics -- Periodicals; 511.6
• DOI: 10.1017/S0963548315000127
• Languages: English
• ISSNs: 0963-5483

Abstract

Abstract : 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.