A CLT FOR INFORMATION-THEORETIC STATISTICS OF GRAM RANDOM MATRICES WITH A GIVEN VARIANCE PROFILE

Consider an N x n random matrix Y-n = (Y-ij(n)) with entries given by Y-ij(n) = sigma(ij)(n)/root n X-ij(n), the X-ij(n) being centered, independent and identically distributed random variables with unit variance and (sigma(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 (Y-n Y-n*+ rho I-N), where Y* is the Hermitian adjoint of Y and rho > 0 is an additional parameter. We prove that, when centered and properly resealed, 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 is an element of (0, infinity). 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 X-ij's differs from the fourth moment of a Gaussian random variable. Such a CLT is of interest in the field of wireless communications.