Arak Inequalities for Concentration Functions and the Littlewood-Offord Problem

Abstract

Let $X,X_1,\dots ,X_n$ 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.

21 pages

References in corpus (1)

• Bound for the maximal probability in the Littlewood-Offord problem

Cited by in corpus (7)

• Forward and Reverse Entropy Power Inequalities in Convex Geometry
• Rogozin's convolution inequality for locally compact groups
• Entropy Inequalities for Sums in Prime Cyclic Groups
• Bound for the maximal probability in the Littlewood-Offord problem
• A combinatorial approach to small ball inequalities for sums and differences
• A relative anti-concentration inequality
• A new bound in the Littlewood--Offord problem