Optimal bounds on tail probabilities — a simplified approach

Abstract

Let Xi ^t8_i=1 be independent random variables, assuming values in [0, 1], having a common mean μ, and variances bounded by σ2. Let S _n = Σ ^{n Xi}_i=1 . We give a general and simple method for obtaining asymptotically optimal upper bounds on probabilities of events of the form S _n - E[S _n] ≥ na with explicit dependence on μ and σ2. For general bounded random variables the method yields the Bennett inequality, with a simplified proof. For specific classes of distributions the method can be used to derive bounds that are tighter than those achieved by the Bennett inequality. We demonstrate the power of the method by applying it to the case of symmetric three-point distributions, thus improving previous results for the List Update Problem.