Deterministic equivalents for certain functionals of large random matrices

Abstract

Consider an $N\times n$ random matrix $Y_n=(Y_{ij}^n)$ where the entries are given by $Y_{ij}^n=\frac{{\sigma }_{ij}(n)}{\sqrt{n}}{X}_{ij}^n$, the $X_{ij}^n$ being independent and identically distributed, centered with unit variance and satisfying some mild moment assumption. Consider now a deterministic $N\times n$ matrix A_n whose columns and rows are uniformly bounded in the Euclidean norm. Let ${\Sigma }_n=Y_n+A_n$. We prove in this article that there exists a deterministic $N\times N$ matrix-valued function T_n(z) analytic in $\mathbb{C}-{\mathbb{R}}^{+}$ such that, almost surely, \[\lim_{n\to+\infty,N/n\to c}\biggl(\frac{1}{N}\operatorname {Trace}(Σ_nΣ_n^T-zI_N)^{-1}-\frac{1}{N}\operatorname {Trace}T_n(z)\biggr)=0.\] Otherwise stated, there exists a deterministic equivalent to the empirical Stieltjes transform of the distribution of the eigenvalues of ${\Sigma }_n{\Sigma }_n^T$. 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 $\frac{1}{N}\mathrm{Trace}{T}_n(z)$ is the Stieltjes transform of a probability measure ${\pi }_n(d\lambda )$, and that for every bounded continuous function f, the following convergence holds almost surely \[\frac{1}{N}\sum_{k=1}^Nf(λ_k)-\int_0^{\infty}f(λ)π_n(dλ)\mathop {\longrightarrow}_{n\to\infty}0,\] where the $({\lambda }_k{)}_{1\le k\le N}$ are the eigenvalues of ${\Sigma }_n{\Sigma }_n^T$. 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: \[C_n(σ^2)=\frac{1}{N}\mathbb{E}\log \det\biggl(I_N+\frac{Σ_nΣ_n^T}{σ^2}\biggr),\] where ${\sigma }^2$ is a known parameter.

Published at http://dx.doi.org/10.1214/105051606000000925 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)