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Central limit theorem for traces of large random symmetric matrices with independent matrix elements. (English) Zbl 0912.15027
The Wigner ensemble of real symmetric random matrices $$A_N(x,y) =a(x,y) N^{-1/2}$$, $$x,y=1, \dots, N$$ is studied in the case when $$\{a(x,y), x\leq y\}$$ are jointly independent random variables. Assuming that the odd moments of $$a(x,y)$$ vanish and the even ones are finite, the authors study asymptotic behaviour of the averaged moments $$\mathbb{E} \text{Tr} A_N^k$$ and the random variable $\xi_{N,k} (t)= \text{Tr} A_N^{kt}-\mathbb{E} \text{Tr} A_N^{kt}$ in the limit when $$N,k \to\infty$$, $$k=o (N^{1/2})$$. The most important result is that $$\xi_{N,k} (t)$$ converges to the Gaussian random variables whose distribution does not depend on the details of the probability distribution of $$\{a(x,y)\}$$. This result represents a rigorous proof of a version of the universality conjecture for large random matrices.

##### MSC:
 15B52 Random matrices (algebraic aspects) 60G50 Sums of independent random variables; random walks 60F05 Central limit and other weak theorems
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##### References:
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