A quantitative central limit theorem for linear statistics of random matrix eigenvalues

TL;DR

This paper establishes a quantitative central limit theorem for linear eigenvalue statistics of Haar-distributed matrices, providing explicit convergence rates under smooth test functions.

Contribution

It introduces a rate of convergence of nearly 1/n for linear eigenvalue statistics, utilizing a recent multivariate CLT and Stein's method.

Findings

Achieves a nearly 1/n convergence rate for eigenvalue statistics.

Applies Stein's method to improve understanding of eigenvalue fluctuations.

Provides quantitative bounds for the CLT in random matrix theory.

Abstract

It is known that the fluctuations of suitable linear statistics of Haar distributed elements of the compact classical groups satisfy a central limit theorem. We show that if the corresponding test functions are sufficiently smooth, a rate of convergence of order almost $1/n$ can be obtained using a quantitative multivariate CLT for traces of powers that was recently proven using Stein's method of exchangeable pairs.