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We study the normalized trace $g_n(z)=n^{-1} \mbox{tr} \, (H-zI)^{-1}$ of theresolvent of $n\times n$ real symmetric matrices$H=\big[(1+\delta_{jk})W_{jk}/\sqrt n\big]_{j,k=1}^n$ assuming that theirentries are independent but not necessarily identically distributed randomvariables. We develop a rigorous method of asymptotic analysis of moments of$g_n(z)$ for $|\Im z| \ge \eta_0$ where $\eta_0$ is determined by the secondmoment of $W_{jk}$. By using this method we find the asymptotic form of theexpectation ${\bf E}\{g_n(z)\}$ and of the connected correlator ${\bfE}\{g_n(z_1)g_n(z_2)\}- {\bf E}\{g_n(z_1)\} {\bf E}\{g_n(z_2)\}$. We also provethat the centralized trace $ng_n(z)- {\bf E}\{ng_n(z)\}$ has the Gaussiandistribution in the limit $n=\infty $. Basing on these results we presentheuristic arguments supporting the universality property of the localeigenvalue statistics for this class of random matrix ensembles.Comment: 29 pages, LaTeX (revtex style files required) submitted to Journ Math  Phy

Asymptotic properties of large random matrices with independent entries