A CLT FOR INFORMATION-THEORETIC STATISTICS OF NON-CENTERED GRAM RANDOM MATRICES

Abstract

In this article, we study the fluctuations of the random variable: [Formula: see text] where [Formula: see text], as the dimensions of the matrices go to infinity at the same pace. Matrices Xn and An are respectively random and deterministic N × n matrices; matrices Dn and [Formula: see text] are deterministic and diagonal, with respective dimensions N × N and n × n; matrix Xn = (Xij) 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 [Formula: see text] satisfies a Central Limit Theorem and has a Gaussian limit. The variance of [Formula: see text] depends on the moment [Formula: see text] of the variables Xij and also on its fourth cumulant [Formula: see text]. The main motivation comes from the field of wireless communications, where [Formula: see text] 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", 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.