98-647 Sinai Ya., Soshnikov A.

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