On the Rate of Decay of the Concentration Function of the Sum of Independent Random Variables

Abstract. Let X1,..., Xn be i.i.d. integral valued random variables and Sn their sum. In the case when X1 has a moderately large tail of distribution, Deshouillers, Freiman and Yudin gave a uniform upper bound for maxk∈Z Pr{Sn = k} (which can be expressed in term of the Lévy Doeblin concentration of Sn), under the extra condition that X1 is not essentially supported by an arithmetic progression. The first aim of the paper is to show that this extra condition cannot be simply ruled out. Secondly, it is shown that if X1 has a very large tail (larger than a Cauchy-type distribution), then the extra arithmetic condition is not sufficient to guarantee a uniform upper bound for the decay of the concentration of the sum Sn. Proofs are constructive and enhance the connection between additive number theory and probability theory