A CLT for Information-theoretic statistics of Gram random matrices with a given variance profile

Consider a N × n random matrix Yn = (Y n ij ) where the entries are given by Y n ij = σij(n) √ n X ij , the X ij being centered, independent and identically distributed random variables with unit variance and (σij(n); 1 ≤ i ≤ N, 1 ≤ j ≤ n) being an array of numbers we shall refer to as a variance profile. We study in this article the fluctuations of the random variable log det (YnY ∗ n + ρIN ) where Y ∗ is the Hermitian adjoint of Y and ρ > 0 is an additional parameter. We prove that when centered and properly rescaled, this random variable satisfies a Central Limit Theorem (CLT) and has a Gaussian limit whose parameters are identified. A complete description of the scaling parameter is given; in particular it is shown that an additional term appears in this parameter in the case where the 4 moment of the Xij ’s differs from the 4 moment of a Gaussian random variable. Such a CLT is of interest in the field of wireless communications.