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

Abstract

In this article, we study the fluctuations of the random variable: $${\mathcal{I}}_n(\rho )=\frac{1}{N}\log \det ({\Sigma }_n{\Sigma }_n^{\ast }+\rho I_N),(\rho >0)$$ 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 $κ= \E|X_{ij}|^4 - 2 - |\E X_{ij}^2|^2$. 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.