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(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 (λk)1≤k≤N 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.

Walid Hachem

Philippe Loubaton

Jamal Najim

The Annals of Applied Probability

应用概率年鉴旨在发表反映当代应用概率各个方面的最高质量研究。主要强调重要性和独创性。

The Annals of Applied Probability aims to publish high-quality research that reflects various aspects of contemporary applied probability. The journal places a strong emphasis on the significance and originality of the research.

《Annals of Applied Probability》旨在发表高质量的研究成果，涵盖当代应用概率的各个方面。期刊特别强调研究的重要性和独创性。

Annals of Applied Probability

Deterministic equivalents for certain functionals of large random matrices

It is proved 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 Sti Beltjestransform of a probability measure.

Category	Quartile
数学	2
数学, 统计学与概率论	2

Deterministic equivalents for certain functionals of large random matrices

ivySCI AI Smartly Parses PDF, Answers Researchers' Questions, and Helps You Understand Papers in Seconds

Journal Info

Category	Quartile
STATISTICS & PROBABILITY	2