A Clt for Information-theoretic Statistics of Gram Random Matrices with a given Variance Profile1 by Walid Hachem,

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. In this article, we study 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 whenever N goes to infinity and Nn → c ∈ (0,∞). 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 fourth moment of the Xij ’s differs from the fourth moment of a Gaussian random variable. Such a CLT is of interest in the field of wireless communications.