## Abstract

Consider an *N*×*n* random matrix *Y*_{n}=(*Y*^{n}_{ij}) where the entries are given by $Y^{n}_{ij}=\frac{\sigma_{ij}(n)}{\sqrt{n}}X^{n}_{ij}$, the *X*^{n}_{ij} being independent and identically distributed, centered with unit variance and satisfying some mild moment assumption. Consider now a deterministic *N*×*n* matrix *A*_{n} whose columns and rows are uniformly bounded in the Euclidean norm. Let Σ_{n}=*Y*_{n}+*A*_{n}. We prove in this article that there exists a deterministic *N*×*N* matrix-valued function *T*_{n}(*z*) analytic in ℂ−ℝ^{+} such that, almost surely, $$\lim_{n\rightarrow+\infty,N/n\rightarrow c}\biggl(\frac{1}{N}\operatorname{Trace}(\Sigma_{n}\Sigma_{n}^{T}-zI_{N})^{-1}-\frac{1}{N}\operatorname{Trace}T_{n}(z)\biggr )=0.$$ Otherwise stated, there exists a deterministic equivalent to the empirical Stieltjes transform of the distribution of the eigenvalues of Σ_{n}Σ_{n}^{T}. For each *n*, the entries of matrix *T*_{n}(*z*) are defined as the unique solutions of a certain system of nonlinear functional equations. It is also proved that $\frac{1}{N}\operatorname{Trace}\ T_{n}(z)$ is the Stieltjes transform of a probability measure *π*_{n}(*dλ*), and that for every bounded continuous function *f*, the following convergence holds almost surely $$\frac{1}{N}\sum_{k=1}^{N}f(\lambda_{k})-\int_{0}^{\infty}f(\lambda)\pi _{n}(d\lambda)\mathop{\longrightarrow}_{n\rightarrow\infty}0,$$ where the (*λ*_{k})_{1≤k≤N} are the eigenvalues of Σ_{n}Σ_{n}^{T}. 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: $$C_{n}(\sigma^{2})=\frac{1}{N}\mathbb{E}\log \det\biggl(I_{N}+\frac{\Sigma_{n}\Sigma_{n}^{T}}{\sigma^{2}}\biggr),$$ where *σ*^{2} is a known parameter.

## Citation

Walid Hachem. Philippe Loubaton. Jamal Najim. "Deterministic equivalents for certain functionals of large random matrices." Ann. Appl. Probab. 17 (3) 875 - 930, June 2007. https://doi.org/10.1214/105051606000000925

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