A CLT for Information-theoretic statistics of Non-centered Gram random matrices

In this article, we study the fluctuations of the random variable: $$ {\mathcal I}_n(\rho) = \frac 1N \log\det(\Sigma_n \Sigma_n^* + \rho I_N),\quad (\rho>0) $$ where \(\Sigma_n= n^{-1/2} D_n^{1/2} X_n\tilde D_n^{1/2} +A_n\), as the dimensions of the matrices go to infinity at the same pace. Matrices \(X_n\) and \(A_n\) are respectively random and deterministic \(N\times n\) matrices; matrices \(D_n\) and \(\tilde D_n\) are deterministic and diagonal, with respective dimensions \(N\times N\) and \(n\times n\); matrix \(X_n=(X_{ij})\) has centered, independent and identically distributed entries with unit variance, either real or complex. We prove that when centered and properly rescaled, the random variable \({\mathcal I}_n(\rho)\) satisfies a Central Limit Theorem and has a Gaussian limit. The variance of \({\mathcal I}_n(\rho)\) depends on the moment \(\E X_{ij}^2\) of the variables \(X_{ij}\) and also on its fourth cumulant \(\kappa= \E|X_{ij}|^4 - 2 - |\E X_{ij}^2|^2\). The main motivation comes from the field of wireless communications, where \({\mathcal I}_n(\rho)\) represents the mutual information of a multiple antenna radio channel. This article closely follows the companion article "A CLT for Information-theoretic statistics of Gram random matrices with a given variance profile", {\em Ann. Appl. Probab. (2008)} by Hachem et al., however the study of the fluctuations associated to non-centered large random matrices raises specific issues, which are addressed here.