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

Abstract

We study Wigner ensembles of symmetric random matricesA=(a _ij ),i, j=1,...,n with matrix elementsa _ij ,i≤j being independent symmetrically distributed random variables

$$a_{ij} = a_{ji} = \frac{{\xi _{ij} }}{{n^{\tfrac{1}{2}} }}.$$

We assume that Var$\xi _{ij} = \frac{1}{4}$, fori<j, Var ξ _ij ≤ const and that all higher moments of ξ _ij also exist and grow not faster than the Gaussian ones. Under formulated conditions we prove the central limit theorem for the traces of powers ofA growing withn more slowly than$\sqrt n$. The limit of Var (TraceA ^p),$1 \ll p \ll \sqrt n$, does not depend on the fourth and higher moments of ξ _ij and the rate of growth ofp, and equals to$\frac{1}{\pi }$. 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.