Central limit theorem for traces of large random symmetric matrices with independent matrix elements

We study Wigner ensembles of symmetric random matrices A = (a_ij), i, j = 1,... , n with matrix elements a_ij, i ≤ j being independent symmetrically distributed random variables aij = aji = ξij/n1/2. We assume that Var ξij = 1/4, for i < j, Var ξii ≤const and that all higher moments of ξij also exist and grow not faster than the Gaussian ones. Under formulated conditions we prove the central limit theorem for the traces of powers of A growing with n more slowly than √n. The limit of Var(Trace A^p), l ≪ p ≪ √n, does not depend on the fourth and higher moments of ξij and the rate of growth of p, and equals to 1/π. As a corollary we improve the estimates on the rate of convergence of the maximal eigenvalue to 1 and prove central limit theorem for a general class of linear statistics of the spectra.