Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices

TL;DR

This paper proves a Central Limit Theorem for linear eigenvalue statistics of Wigner and sample covariance matrices under weaker conditions than previously known, introducing a universal method for variance bounds applicable to various ensembles.

Contribution

It establishes CLTs under weaker assumptions and develops a universal approach for variance bounds in random matrix theory.

Findings

CLT proven for Wigner and covariance matrices under weak conditions

Universal method for variance bounds based on resolvent trace

Applicable to a broad class of random matrix ensembles

Abstract

We consider two classical ensembles of the random matrix theory: the Wigner matrices and sample covariance matrices, and prove Central Limit Theorem for linear eigenvalue statistics under rather weak (comparing with results known before) conditions on the number of derivatives of the test functions and also on the number of the entries moments. Moreover, we develop a universal method which allows one to obtain automatically the bounds for the variance of differentiable test functions, if there is a bound for the variance of the trace of the resolvent of random matrix. The method is applicable not only to the Wigner and sample covariance matrices, but to any ensemble of random matrices.