A CLT for information-theoretic statistics of non-centered Gram random matrices

In(ρ) = 1 N log det (ΣnΣ ∗ n + ρIN ) , (ρ > 0) where Σn = n−1/2D 1/2 n XnD̃ 1/2 n + An, as the dimensions of the matrices go to infinity at the same pace. Matrices Xn and An are respectively random and deterministic N ×n matrices; matrices Dn and D̃n are deterministic and diagonal, with respective dimensions N×N and n×n; matrixXn = (Xij) has centered, independent and identically distributed entries with unit variance, either real or complex. We prove that when centered and properly rescaled, the random variable In(ρ) satisfies a Central Limit Theorem and has a Gaussian limit. The variance of In(ρ) depends on the moment EX ij of the variables Xij and also on its fourth cumulant κ = E|Xij | 4 − 2− |EX ij | . The main motivation comes from the field of wireless communications, where In(ρ) represents the mutual information of a multiple antenna radio channel. This article closely follows the companion article ”A CLT for Information-theoretic statistics of Gram random matrices with a given variance profile”, Ann. Appl. Probab. (2008) by Hachem et al., however the study of the fluctuations associated to non-centered large random matrices raises specific issues, which are addressed here.