In a celebrated work by Hoeffding [J. Amer. Statist. Assoc. 58 (1963) 13-30], several inequalities for tail probabilities of sums M_n=X_1+... +X_n of bounded independent random variables X_j 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 P{M_n\geq x}\leq cP{S_n\geq x}, where c is an absolute constant and S_n=ε_1+... +ε_n 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\in R where the survival function x\mapsto P{S_n\geq 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 P{S_n\geq 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.

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