2004/10/06 by Vidmantas Bentkus
Mathematics · #math.PR #msc:60E15
paper · pdf · doi:10.1214/009117904000000360
published as Annals of Probability 2004, Vol. 32, No. 2, 1650-1673 · Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000360
arxiv created 2004/10/06 · arxiv updated 2009/12/01
In a celebrated work by Hoeffding [J. Amer. Statist. Assoc. 58 (1963) 13-30], several inequalities for tail probabilities of sums Mn=X1+... +Xn of bounded independent random variables Xj were proved. These inequalities had a considerable impact on the development of probability and statistics, and remained unimproved until 1995 when Talagrand [Inst. Hautes Etudes Sci. Publ. Math. 81 (1995a) 73-205] inserted certain missing factors in the bounds of two theorems. By similar factors, a third theorem was refined by Pinelis [Progress in Probability 43 (1998) 257-314] and refined (and extended) by me. In this article, I introduce a new type of inequality. Namely, I show that PMn≥ x≤ cPSn≥ x, where c is an absolute constant and Sn=ε1+... +εn is a sum of independent identically distributed Bernoulli random variables (a random variable is called Bernoulli if it assumes at most two values). The inequality holds for those x∈ R where the survival function x↦ PSn≥ x has a jump down. For the remaining x the inequality still holds provided that the function between the adjacent jump points is interpolated linearly or log-linearly. If it is necessary, to estimate PSn≥ x special bounds can be used for binomial probabilities. The results extend to martingales with bounded differences. It is apparent that Theorem 1.1 of this article is the most important.