On the rate of convergence to the semi-circular law

TL;DR

This paper establishes that the empirical spectral distribution of certain Hermitian random matrices converges to the semi-circular law at a rate of order $O(n^{-1} ext{log}^b n)$, under sub-exponential decay conditions on matrix entries.

Contribution

It provides a new rate of convergence result for the spectral distribution of Wigner matrices with sub-exponential decay of entries.

Findings

Convergence rate of $O(n^{-1} ext{log}^b n)$ for spectral distribution

Applicable to matrices with sub-exponential tail decay

Uses recursion argument for proof

Abstract

Let $\mathbf X=(X_{jk})$ denote a Hermitian random matrix with entries $X_{jk}$, which are independent for $1\le j\le k$. We consider the rate of convergence of the empirical spectral distribution function of the matrix $\mathbf X$ to the semi-circular law assuming that $\mathbf E X_{jk}=0$, $\mathbf E X_{jk}^2=1$ and that the distributions of the matrix elements $X_{jk}$ have a uniform sub exponential decay in the sense that there exists a constant $\varkappa>0$ such that for any $1\le j\le k\le n$ and any $t\ge 1$ we have $$ \Pr\{|X_{jk}|>t\}\le \varkappa^{-1}\exp\{-t^{\varkappa}\}. $$ By means of a recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the Wigner matrix $\mathbf W=\frac1{\sqrt n}\mathbf X$ and the semicircular law is of order $O(n^{-1}\log^b n)$ with some positive constant $b>0$.