Multivariate estimates for the concentration functions of weighted sums of independent identically distributed random variables

TL;DR

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

Contribution

It formulates and proves multidimensional generalizations of existing results on concentration functions, refining previous bounds and extending their applicability.

Findings

Multidimensional generalizations of concentration function estimates

Refined bounds for eigenvalue distribution analysis

Enhanced understanding of weighted sum behaviors

Abstract

Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. The paper deals with the question about the behavior of the concentration function of the random variable $\sum\limits_{k=1}^{n}X_k a_k$ according to the arithmetic structure of vectors $a_k$. 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 multidimensional generalizations of the results Eliseeva and Zaitsev (2012). They are also the refinements of the results of Friedland and Sodin (2007) and Rudelson and Vershynin (2009).