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Limit theorems for random matrices. (English) Zbl 0917.60040
The authors consider ensembles of random real symmetric \(N\times N\) matrices \(H_N\) whose entries are weakly dependent Gaussian random variables with the covariance matrix \(V\). It is proved that if \(g_N(z)\) is the Stieltjes transform of the normalized eigenvalue counting function \(\sigma (\lambda;H_N)\), and \(|\text{Im} z|\geq w_0\) for a positive and \(N\)-independent \(w_0\), then the centered random variables \(N(g_N(z) -Eg_N(z))\) converge in distribution to a Gaussian random variable as \(N\to\infty\).

60F99 Limit theorems in probability theory
15B52 Random matrices (algebraic aspects)
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