On the Central Limit Theorem for the Eigenvalue Counting Function of Wigner and Covariance matrices

TL;DR

This paper establishes central limit theorems for the eigenvalue counting function of Wigner and covariance matrices, extending known results to broader classes of matrices using the Four Moment Theorem.

Contribution

It extends central limit theorems for eigenvalue counts to large classes of Wigner and covariance matrices via the Four Moment Theorem.

Findings

CLTs for eigenvalue counting functions in Wigner matrices

Extension of results to covariance matrices

Application of Four Moment Theorem for broader classes

Abstract

This note presents some central limit theorems for the eigenvalue counting function of Wigner matrices in the form of suitable translations of results by Gustavsson and O'Rourke on the limiting behavior of eigenvalues inside the bulk of the semicircle law for Gaussian matrices. The theorems are then extended to large families of Wigner matrices by the Tao and Vu Four Moment Theorem. Similar results are developed for covariance matrices.