Central limit theorem and convergence of the support for Wishart matrices with correlated entries

Abstract: In this article, we will consider Wishart Matrices with correlated entries, but with a strictly log-concave law. It has been shown by A.Pajor and L.Pastur that the empirical measures of such matrices converges. We will show, under some symmetry and convergence hypotheses, that in this case the central limit theorem for linear statistics holds around their limit, and deduce the convergence of the support to the support of the limiting measure. We also provide the convergence of the Expectation and the Covariance.