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

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 $n^{-1}{X}^{\ast }X$ 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.

Published in at http://dx.doi.org/10.1214/09-AOP452 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Gaussian Fluctuation in Random Matrices
• Limiting Laws of Linear Eigenvalue Statistics for Unitary Invariant Matrix Models
• Bulk Universality and Related Properties of Hermitian Matrix Models

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