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

Abstract

Consider an N×n random matrix Y_n=(Yⁿ_ij) with entries given by $$Y_{ij}^{n}=\frac{\sigma_{ij}(n)}{\sqrt{n}}X_{ij}^{n},$$ the Xⁿ_ij being centered, independent and identically distributed random variables with unit variance and (σ_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_nY^*_n+ρI_N),

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 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 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.