Optimal Hoeffding-like inequalities under a symmetry assumption

In this paper, we prove a Hoeffding-like inequality for the survival function of a sum of symmetric independent identically distributed random variables, taking values in a segment [−b, b] of the reals. The symmetric case is relevant to the auditing practice and is an important case study for further investigations. The bounds as given by Hoeffding in 1963 cannot be improved upon unless we restrict the class of random variables, for instance, by assuming the law of the random variables to be symmetric with respect to their mean, which we may assume to be zero. The main result in this paper is an improvement of the Hoeffding bound for i.i.d. random variables which are bounded and have a (upper bound for the) variance by further assuming that they have a symmetric law.