Sinai Ya., Soshnikov A.
Central Limit Theorem for Traces of Large Random Symmetric Matrices With
Independent Matrix Elements
(555K, PostScript)

ABSTRACT.  We study Wigner ensembles of symmetric random matrices
$A= (a_{ij}) \; i,j = 1, \ldots , n$ with matrix elements
$a_{ij} ,  \quad  i\leq j$ being independent symmetrically distributed
random variables
$$
a_{ij}= \frac{\xi_(ij}}{n^{\frac{1}{2}}}
$$
such that $Var(\xi_{ij})= \frac{1}{4}$
for $i<j$,  and all higher moments of $\xi_{ij}$ also exist and grow not
faster than the Gaussian ones.
Under formulated conditions we prove the central limit theorem for the
traces of powers of $A$ growing with $n$ more slowly than 
$\sqrt{n}$. The limit of $ Var( Trace A^p), \; 1 \ll p \ll \sqrt{n}$
does not depend on the fourth and higher moments of $\xi_{ij}$ and the
rate of growth of $p$, 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 spectrum.