Math Motivation

THE RAMANUJAN JOURNAL, 9, 241–250, 2005
c @2005 Springer Science + Business Media, Inc. Manufactured in the Netherlands.

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

JEAN-MARC DESHOUILLERS jean-marc.deshouillers@math.u-bordeaux1.fr
Statistique Mathematique et Applications, EA 2961, Universite Victor Segalen Bordeaux 2, Bordeaux, France

SUTANTO sutanto@uns.ac.id
Universitas Sebalas Maret, Surakarta, Indonesia

A Jean-Louis Nicolas, avec amiti´e et respect
Received May 1, 2003; Accepted January 28, 2005

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 Levy 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 showthat 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.

Key words: Levy concentration, sum of i.i.d. random variables, arithmetic progression, additive number theory

2000 Mathematics Subject Classification: Primary—60Fxx, 60Exx, 11Pxx, 11B25

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