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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 "Let us estimate the characterictic function H π,1 (t) for  t ≤ D. We can proceed in the same way as the authors of [9], [18] and [21]. It is evident that 1 − cos x ≥ 2x 2 /π 2 , for x ≤ π. "
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ABSTRACT: Let $X,X_1,...,X_n$ be independent identically distributed random variables. In this paper we study the behavior of the concentration functions of the weighted sums $\sum\limits_{k=1}^{n}a_k X_k$ with respect to the arithmetic structure of coefficients $a_k$. Such concentration results recently became important in connection with investigations about singular values of random matrices. In this paper we formulate and prove some refinements of a result of Vershynin (R. Vershynin, Invertibility of symmetric random matrices, arXiv:1102.0300. (2011). Published in Random Structures and Algorithms, v. 44, no. 2, 135182 (2014)). 
 "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. "
Article: On the LittlewoodOfford problem
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ABSTRACT: The paper deals with studying a connection of the LittlewoodOfford problem with estimating the concentration functions of some symmetric infinitely divisible distributions. Some multivariate generalizations of results of Arak (1980) are given. They show a connection of the concentration function of the sum with the arithmetic structure of supports of distributions of independent random vectors for arbitrary distributions of summands. 
 "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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ABSTRACT: Let $X,X_1,...,X_n$ be independent identically distributed random variables. The paper deals with the question about the behavior of the concentration function of the random variable $\sum_{k=1}^{n}a_k X_k$ according to the arithmetic structure of coefficients $a_k$. Recently the interest to this question has increased significantly due to the study of distributions of eigenvalues of random matrices. In this paper we formulate and prove some refinements of the results of Friedland and Sodin (2007) and Rudelson and Vershynin (2009).