## The Annals of Applied Probability

### A CLT for information-theoretic statistics of Gram random matrices with a given variance profile

#### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 18, Number 6 (2008), 2071-2130.

Dates
First available in Project Euclid: 26 November 2008

https://projecteuclid.org/euclid.aoap/1227708913

Digital Object Identifier
doi:10.1214/08-AAP515

Mathematical Reviews number (MathSciNet)
MR2473651

Zentralblatt MATH identifier
1166.15013

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

#### Citation

Hachem, Walid; Loubaton, Philippe; Najim, Jamal. A CLT for information-theoretic statistics of Gram random matrices with a given variance profile. Ann. Appl. Probab. 18 (2008), no. 6, 2071--2130. doi:10.1214/08-AAP515. https://projecteuclid.org/euclid.aoap/1227708913

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