Estimates for the concentration functions in the Littlewood--Offord problem

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 $\sum _{k=1}^n{a}_k{X}_k$ 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)).

16 pages

References in corpus (3)

• Estimates for the concentration functions in the Littlewood--Offord problem
• Estimates for the concentration functions of weighted sums of independent random variables
• Bounds on the concentration function in terms of Diophantine approximation

Cited by in corpus (5)

• Estimates for the concentration functions in the Littlewood--Offord problem
• Arak Inequalities for Concentration Functions and the Littlewood-Offord Problem
• Bound for the maximal probability in the Littlewood-Offord problem
• On the Littlewood--Offord problem
• Arak Inequalities for Concentration Functions and the Littlewood--Offord Problem: a shortened version