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

Consider an N.n random matrix Yn=(Ynij) with entries given by Ynij=.ij(n).nXnij, the Xnij being centered, independent and identically distributed random variables with unit variance and (.ij(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+.IN), where Y* is the Hermitian adjoint of Y and .>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 N/n.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.