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

Let X₁, ... , X_n be i.i.d. integral valued random variables and S_n their sum. In the case when X₁ has a moderately large tail of distribution, Deshouillers, Freiman and Yudin gave a uniform upper bound for max _{k ∊ ℤ} Pr{S_n = k} (which can be expressed in term of the Lévy Doeblin concentration of S_n), 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 S_n. Proofs are constructive and enhance the connection between additive number theory and probability theory.À Jean-Louis Nicolas, avec amitié et respect2000 Mathematics Subject Classification: Primary—60Fxx, 60Exx, 11Pxx, 11B25