Eigenvalue distribution of large random matrices with correlated entries

We study the normalized eigenvalue counting function $N_n(\lambda)$ of an ensemble of $n\times n$ symmetric random matrices with statistically dependent arbitrary distributed entries $u_n(x,y)$, $x,y=1,\dots,n$. We prove that if the correlation function $S$ of the entries is the same for each $n$ and the correlation coefficient of random fields $\{u_n(x,y)\}$ decays fast enough, then in the limit $n\to\infty$ the measure $N_n(d\lambda)$ weakly converges in probability to a nonrandom measure $N(d\lambda)$. We derive an equation for the Stieltjes transform of limiting $N_n(d\lambda)$ and show that the latter depends only on the limiting matrix of averages of $u_n(x,y)$ and the correlation function $S$.