Central limit theorem for linear eigenvalue statistics of random matrices with independent entries

TL;DR

This paper proves a central limit theorem for linear eigenvalue statistics of certain random matrices with independent entries, extending known results to non-Gaussian cases with zero and nonzero excess kurtosis.

Contribution

It introduces a simple interpolation method to establish CLTs for eigenvalue statistics of Wigner and sample covariance matrices with non-Gaussian entries, including nonzero excess kurtosis cases.

Findings

CLT holds for eigenvalue statistics with Gaussian-like variance.

Extension of CLT to matrices with nonzero excess kurtosis.

Variance includes an additional term proportional to the excess kurtosis.

Abstract

We consider $n\times n$ real symmetric and Hermitian Wigner random matrices $n^{-1/2}W$ with independent (modulo symmetry condition) entries and the (null) sample covariance matrices $n^{-1}X^*X$ with independent entries of $m\times n$ matrix $X$. Assuming first that the 4th cumulant (excess) $\kappa_4$ of entries of $W$ and $X$ is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as $n\to\infty$, $m\to\infty$, $m/n\to c\in[0,\infty)$ with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class $\mathbf{C}^5$). This is done by using a simple ``interpolation trick'' from the known results for the Gaussian matrices and the integration by parts, presented in the form of certain differentiation formulas. Then, by using a more elaborated version of the techniques, we prove the CLT in the case of nonzero excess of entries again for essentially $\mathbb{C}^5$ test function. Here the variance of statistics contains an additional term proportional to $\kappa_4$. The proofs of all limit theorems follow essentially the same scheme.