Article

# Bounds on the concentration function in terms of Diophantine approximation

School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
(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

### Full-text preview

Available from: ArXiv
• Source
• "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. "
##### Article: Estimates for the Concentration Functions in the Littlewood–Offord Problem
[Hide abstract]
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, 135--182 (2014)).
Full-text · Article · Mar 2015 · Journal of Mathematical Sciences
• Source
• "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 Littlewood-Offord problem
[Hide abstract]
ABSTRACT: The paper deals with studying a connection of the Littlewood-Offord 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.
Full-text · Article · Nov 2014
• Source
• "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] "
##### Article: Estimates of the concentration functions of weighted sums of independent random variables
[Hide abstract]
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).
Full-text · Article · Mar 2013 · Theory of Probability and Its Applications