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

In this article, we study the fluctuations of the random variable: I-n(rho) = 1/N log det(Sigma(n)Sigma(*)(n) +.rho I-N), (rho > 0) where Sigma(n) = n(-1/2) D-n(1/2) X-n(D) over tilde (1/2)(n) + 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 x n matrices; matrices D-n and (D) over tilde (n) are deterministic and diagonal, with respective dimensions N x N and n x 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 In(rho) satisfies a Central Limit Theorem and has a Gaussian limit. The variance of In(rho) depends on the moment EXij2 of the variables X-ij and also on its fourth cumulant kappa = E|X-ij|(4) - 2 - |EXij2|(2). The main motivation comes from the field of wireless communications, where In(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", 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.