Deterministic equivalents for certain functionals of large random matrices

Consider a $N\times n$ random matrix $ Y_n$ where the entries are independent but not identically distributed (matrices with a variance profile) Consider now a deterministic $N\times n$ matrix $A_n$ whose columns and rows are uniformly bounded for the Euclidean norm. Let $\Sigma_n=Y_n+A_n$. We prove in this article that there exists a deterministic equivalent to the empirical Stieltjes transform of the distribution of the eigenvalues of $\Sigma_n \Sigma_n^T$ which is itself the Stieltjes transform of a probability measure. This work is motivated by the context of performance evaluation of Multiple Inputs / Multiple Output (MIMO) wireless digital communication channels. As an application, we derive a deterministic equivalent to the mutual information of a wireless channel.

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