On the Bernstein-Hoeffding method

Abstract

We show that the Bernstein-Hoeffding method can be employed to a larger class of generalized moments. This class includes the exponential moments whose properties play a key role in the proof of a well-known inequality of Wassily Hoeffding, for sums of independent and bounded random variables whose mean is assumed to be known. As a result we can generalise and improve upon this inequality. We show that Hoeffding's bound is optimal in a broader sense. Our approach allows to obtain "missing" factors in Hoeffding's inequality whose existence is motivated by the central limit theorem. The later result is a rather weaker version of a theorem that is due to Michel Talagrand. Using ideas from the theory of Bernstein polynomials, we show that the Bernstein-Hoeffding method can be adapted to case in which one has information on higher moments of the random variables. Moreover, we consider the performance of the method under additional information on the conditional distribution of the random variables and, finally, we show that the method reduces to Markov's inequality when employed to non-negative and unbounded random variables.