Article

Bounds on the concentration function in terms of Diophantine approximation

School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
Comptes Rendus Mathematique (Impact Factor: 0.47). 07/2007; 345(9). DOI: 10.1016/j.crma.2007.10.006
Source: arXiv

ABSTRACT

We demonstrate a simple analytic argument that may be used to bound the Lévy concentration function of a sum of independent random variables. The main application is a version of a recent inequality due to Rudelson and Vershynin, and its multidimensional generalisation. Des bornes pour la fonction de concentration en matière d’approximation Diophantienne. Nous montrons un simple raisonnement analytique qui peut être utile pour borner la fonction de concentration d’une somme des variables aléatoires indépendantes. L’application principale est une version de l’inégalité récente de Rudelson et Vershynin, et sa généralisation au cadre multidimensionel. 1

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    • "In the same paper there is a proof of multidimensional analogs of some results of Arak [1]. In Theorems 2 and 3 below, we provide without proof the formulations of these results which demonstrates a relation between the order of smallness of the concentration function of the sum and the arithmetic structure of the supports of distributions of independent random vectors for arbitrary distributions of summands, in contrast to the results of [9], [13], [16]–[20], in which a similar relationship was found in a particular case of summands with the distributions arising in the Littlewood–Offord problem. We need some notation. "
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    • "The proofs of our Theorem 1 and Corollary 1 are in some sence easier than the proofs in Friedland and Sodin [6] and Rudelson and Vershynin [15] "
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