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

Abstract

We consider n×n real symmetric and Hermitian Wigner random matrices n^−1/2W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n⁻¹X^*X with independent entries of m×n matrix X. Assuming first that the 4th cumulant (excess) κ₄ 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→∞, m→∞, m/n→c∈[0, ∞) with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class C⁵). 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 ℂ⁵ test function. Here the variance of statistics contains an additional term proportional to κ₄. The proofs of all limit theorems follow essentially the same scheme.