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December 2008 A CLT for information-theoretic statistics of Gram random matrices with a given variance profile
Walid Hachem, Philippe Loubaton, Jamal Najim
Ann. Appl. Probab. 18(6): 2071-2130 (December 2008). DOI: 10.1214/08-AAP515

Abstract

Consider an N×n random matrix Yn=(Ynij) with entries given by $$Y_{ij}^{n}=\frac{\sigma_{ij}(n)}{\sqrt{n}}X_{ij}^{n},$$ the Xnij being centered, independent and identically distributed random variables with unit variance and (σij(n); 1≤iN, 1≤jn) being an array of numbers we shall refer to as a variance profile. In this article, we study the fluctuations of the random variable

log det(YnY*n+ρIN),

where Y* is the Hermitian adjoint of Y and ρ>0 is an additional parameter. We prove that, when centered and properly rescaled, this random variable satisfies a central limit theorem (CLT) and has a Gaussian limit whose parameters are identified whenever N goes to infinity and N/nc∈(0, ∞). A complete description of the scaling parameter is given; in particular, it is shown that an additional term appears in this parameter in the case where the fourth moment of the Xij’s differs from the fourth moment of a Gaussian random variable. Such a CLT is of interest in the field of wireless communications.

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Walid Hachem. Philippe Loubaton. Jamal Najim. "A CLT for information-theoretic statistics of Gram random matrices with a given variance profile." Ann. Appl. Probab. 18 (6) 2071 - 2130, December 2008. https://doi.org/10.1214/08-AAP515

Information

Published: December 2008
First available in Project Euclid: 26 November 2008

zbMATH: 1166.15013
MathSciNet: MR2473651
Digital Object Identifier: 10.1214/08-AAP515

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

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

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.18 • No. 6 • December 2008
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