A CLT for information-theoretic statistics of non-centered Gram random matrices (Q2884857)

The authors consider an \(N\times n\) random matrix \(\Sigma_n\) given by \(\Sigma_n=n^{-1/2}D_n^{1/2}X_n\tilde D_n^{1/2}+A_n\), where \(X_n\) and \(A_n\) are, respectively, random and deterministic \(N\times n\) matrices; \(D_n\) and \(\tilde D_n\) are deterministic square diagonal matrices of appropriate dimensions; the matrix \(X_n=(X_{ij})\) has centered, independent and identically distributed entries with unit variance, either real or complex. The authors study the fluctuations of the random variable NEWLINE\[NEWLINE{\mathcal I}_n(\rho)=(1/N)\log\det(\Sigma_n\Sigma_n^*+\rho I_N),\;\rho>0,NEWLINE\]NEWLINE as the dimension \(n\) goes to infinity.NEWLINENEWLINEThey prove that when centered and properly rescaled, the random variable \({\mathcal I}_n(\rho)\) satisfies a central limit theorem (CLT) and has a Gaussian limit.NEWLINENEWLINEThe main motivation comes from the field of wireless communications, where \({\mathcal I}_n(\rho)\) represents the mutual information of a multiple antenna radio channel.