The Annals of Applied Probability

Deterministic equivalents for certain functionals of large random matrices

Walid Hachem, Philippe Loubaton, and Jamal Najim

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Abstract

Consider an N×n random matrix Yn=(Ynij) where the entries are given by $Y^{n}_{ij}=\frac{\sigma_{ij}(n)}{\sqrt{n}}X^{n}_{ij}$, the Xnij being independent and identically distributed, centered with unit variance and satisfying some mild moment assumption. Consider now a deterministic N×n matrix An whose columns and rows are uniformly bounded in the Euclidean norm. Let Σn=Yn+An. We prove in this article that there exists a deterministic N×N matrix-valued function Tn(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 ΣnΣnT. For each n, the entries of matrix Tn(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 πn(), 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 (λk)1≤kN are the eigenvalues of ΣnΣnT. 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 σ2 is a known parameter.

Article information

Source
Ann. Appl. Probab., Volume 17, Number 3 (2007), 875-930.

Dates
First available in Project Euclid: 22 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1179839177

Digital Object Identifier
doi:10.1214/105051606000000925

Mathematical Reviews number (MathSciNet)
MR2326235

Zentralblatt MATH identifier
1181.15043

Subjects
Primary: 15A52
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors 60F15: Strong theorems

Keywords
Random matrix empirical distribution of the eigenvalues Stieltjes transform

Citation

Hachem, Walid; Loubaton, Philippe; Najim, Jamal. Deterministic equivalents for certain functionals of large random matrices. Ann. Appl. Probab. 17 (2007), no. 3, 875--930. doi:10.1214/105051606000000925. https://projecteuclid.org/euclid.aoap/1179839177


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