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

Abstract

Consider an N×n random matrix Yn=(Ynij) where the entries are given by Yijn=σij(n)nXijn, 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, limn+,N/nc(1NTrace(ΣnΣnTzIN)11NTraceTn(z))=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 1NTrace Tn(z) is the Stieltjes transform of a probability measure πn(), and that for every bounded continuous function f, the following convergence holds almost surely 1Nk=1Nf(λk)0f(λ)πn(dλ)n0, 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: where σ2 is a known parameter.

Citation

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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

Information

Published: June 2007
First available in Project Euclid: 22 May 2007

zbMATH: 1181.15043
MathSciNet: MR2326235
Digital Object Identifier: 10.1214/105051606000000925

Subjects:
Primary: 15A52
Secondary: 15A18‎ , 60F15

Keywords: Empirical distribution of the eigenvalues , Random matrix , Stieltjes transform

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.17 • No. 3 • June 2007
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