  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.
