Central limit theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices

We prove central limit theorem for linear eigenvalue statistics of orthogonally invariant ensembles of random matrices with one interval limiting spectrum. We consider ensembles with real analytic potentials and test functions with two bounded derivatives. 1 Introduction and main result In this paper we consider ensembles of n×n real symmetric matrices M with the probability distribution Pn(M)dM = Z−1n,β exp{