# zbMATH — the first resource for mathematics

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 $$N\to\infty$$.

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