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.
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.
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Omer Friedland 1; Sasha Sodin 1
@article{CRMATH_2007__345_9_513_0, author = {Omer Friedland and Sasha Sodin}, title = {Bounds on the concentration function in terms of the {Diophantine} approximation}, journal = {Comptes Rendus. Math\'ematique}, pages = {513--518}, publisher = {Elsevier}, volume = {345}, number = {9}, year = {2007}, doi = {10.1016/j.crma.2007.10.006}, language = {en}, }
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/
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[2] Estimates for the concentration function of combinatorial number theory and probability, Period. Math. Hungar., Volume 8 (1977) no. 3–4, pp. 197-211
[3] Estimates on the concentration function in : Notes on Lectures of Oskolkov http://www.math.sc.edu/~howard/Notes/concentration.pdf
[4] The Littlewood–Offord problem and invertibility of random matrices (arxiv preprint) | arXiv
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