Estimates for the concentration functions of weighted sums of independent random variables

TL;DR

This paper investigates how the concentration function of weighted sums of i.i.d. random variables depends on the coefficients' structure, with implications for understanding eigenvalue distributions in random matrices.

Contribution

It refines existing results on concentration functions for weighted sums, providing new bounds based on the coefficients' arithmetic structure.

Findings

Refined bounds for concentration functions of weighted sums.

Connections established between coefficient structure and eigenvalue distributions.

Enhanced understanding of random matrix eigenvalue behavior.

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).