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We consider $n\times n$ random matrices $M_{n}=\sum_{\alpha =1}^{m}{\tau_{\alpha }}\mathbf{y}_{\alpha }\otimes \mathbf{y}_{\alpha }$, where $\tau_{\alpha }\in \mathbb{R}$, $\{\mathbf{y}_{\alpha }\}_{\alpha =1}^{m}$ arei.i.d. isotropic random vectors of $\mathbb{R}^n$, whose components are notnecessarily independent. It was shown in arXiv:0710.1346 that if$m,n\rightarrow \infty$, $m/n\rightarrow c\in \lbrack 0,\infty )$, theNormalized Counting Measures of $\{\tau _{\alpha }\}_{\alpha =1}^{m}$ convergeweakly and $\{\mathbf{y}_\alpha\}_{\alpha=1}^m$ are \textit{good} (seecorresponding definition), then the Normalized Counting Measures of eigenvaluesof $M_{n}$ converge weakly in probability to a non-random limit found in\cite{Ma-Pa:67}. In this paper we indicate a subclass of good vectors, which wecall \textit{very good} and for which the linear eigenvalue statistics of thecorresponding matrices converge in distribution to the Gaussian law, i.e., theCentral Limit Theorem is valid. An important example of good vectors, studiedin arXiv:0710.1346 are the vectors with log-concave distribution. We discussthe conditions for them, guaranteeing the validity of the Central Limit Theoremfor linear eigenvalue statistics of corresponding matrices.

The Central Limit Theorem for Linear Eigenvalue Statistics of the Sum of Independent Matrices of Rank One