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

We consider \(n\times n\) random matrices \(M_{n}=\sum_{\alpha =1}^{m}{\tau _{\alpha }}\mathbf{y}_{\alpha }\otimes \mathbf{y}_{\alpha }\), where \(\tau _{\alpha }\in \mathbb{R}\), \(\{\mathbf{y}_{\alpha }\}_{\alpha =1}^{m}\) are i.i.d. isotropic random vectors of \(\mathbb{R}^n\), whose components are not necessarily independent. It was shown in arXiv:0710.1346 that if \(m,n\rightarrow \infty\), \(m/n\rightarrow c\in \lbrack 0,\infty )\), the Normalized Counting Measures of \(\{\tau _{\alpha }\}_{\alpha =1}^{m}\) converge weakly and \(\{\mathbf{y}_\alpha\}_{\alpha=1}^m\) are \textit{good} (see corresponding definition), then the Normalized Counting Measures of eigenvalues of \(M_{n}\) 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 we call \textit{very good} and for which the linear eigenvalue statistics of the corresponding matrices converge in distribution to the Gaussian law, i.e., the Central Limit Theorem is valid. An important example of good vectors, studied in arXiv:0710.1346 are the vectors with log-concave distribution. We discuss the conditions for them, guaranteeing the validity of the Central Limit Theorem for linear eigenvalue statistics of corresponding matrices.