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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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