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.
"Central limit theorem for linear eigenvalue statistics of random matrices with independent entries." Ann. Probab. 37 (5) 1778 - 1840, September 2009. https://doi.org/10.1214/09-AOP452