 98647 Sinai Ya., Soshnikov A.
 Central Limit Theorem for Traces of Large Random Symmetric Matrices With
Independent Matrix Elements
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Oct 14, 98

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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.
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