Comptes Rendus
no. 9
Probability Theory
Bounds on the concentration function in terms of the Diophantine approximation
[Des bornes pour la fonction de concentration en matière d'approximation diophantienne]

Présenté par : Jean Bourgain

Omer Friedland1 ; Sasha Sodin1
1 School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
Comptes Rendus. Mathématique, Volume 345 (2007) no. 9, pp. 513-518.

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.

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 generalization.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.10.006
Omer Friedland 1 ; Sasha Sodin 1

1 School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel 
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Omer Friedland; Sasha Sodin. Bounds on the concentration function in terms of the Diophantine approximation. Comptes Rendus. Mathématique, Volume 345 (2007) no. 9, pp. 513-518. doi : 10.1016/j.crma.2007.10.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.006/

[1] H.J. Brascamp; E.H. Lieb; J.M. Luttinger A general rearrangement inequality for multiple integrals, J. Functional Anal., Volume 17 (1974), pp. 227-237

[2] G. Halász Estimates for the concentration function of combinatorial number theory and probability, Period. Math. Hungar., Volume 8 (1977) no. 3–4, pp. 197-211

[3] R. Howard Estimates on the concentration function in Rd: Notes on Lectures of Oskolkov http://www.math.sc.edu/~howard/Notes/concentration.pdf

[4] M. Rudelson; R. Vershynin The Littlewood–Offord problem and invertibility of random matrices (arxiv preprint) | arXiv

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