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: 1.09). 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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    • "is actually the starting point of almost all recent studies on the Littlewood–Offord problem (usually for τ = κ, see, for instance, [19], [24], [32], [37], [38] and [43]). More precisely, with the help of Lemma 8 or its analogues, the authors of the above-mentioned papers have obtained estimates of the type "
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    ABSTRACT: Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. In this paper we study the behavior of concentration functions of weighted sums $\sum_{k=1}^{n}a_k X_k$ with respect to the arithmetic structure of coefficients $a_k$ in the context of the Littlewood--Offord problem. Concentration results of this type received renewed interest in connection with distributions of singular values of random matrices. Recently, Tao and Vu proposed an Inverse Principle in the Littlewood-Offord problem. We discuss the relations between the Inverse Principle of Tao and Vu as well as that of Nguyen and Vu and a similar principle formulated for sums of arbitrary independent random variables in the work of Arak from the 1980's.
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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, 135--182 (2014)).
    Journal of Mathematical Sciences 03/2015; 206(2):146-158. DOI:10.1007/s10958-015-2299-3
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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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    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.
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