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

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Abstract : Consider a $N\times n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are given by $Y_{ij}^{n}=\frac{\sigma_{ij}(n)}{\sqrt{n}} X_{ij}^{n}\ ,$ the $X_{ij}^{n}$ being centered, independent and identically distributed random variables with unit variance and $(\sigma_{ij}(n); 1\le i\le N, 1\le j\le n)$ being an array of numbers we shall refer to as a variance profile. We study in this article the fluctuations of the random variable $\log\det\left(Y_n Y_n^* + \rho I_N \right)$ where $Y^*$ is the Hermitian adjoint of $Y$ and $\rho > 0$ 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. 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 4$^\textrm{th}$ moment of the $X_{ij}$'s differs from the 4$^{\textrm{th}}$ moment of a Gaussian random variable. Such a CLT is of interest in the field of wireless communications.
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https://hal.archives-ouvertes.fr/hal-00151136
Contributor : Walid Hachem <>
Submitted on : Friday, June 1, 2007 - 4:35:53 PM
Last modification on : Wednesday, February 20, 2019 - 2:40:44 PM
Long-term archiving on : Thursday, April 8, 2010 - 6:42:43 PM

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• HAL Id : hal-00151136, version 1
• ARXIV : 0706.0166

### Citation

Walid Hachem, Philippe Loubaton, Jamal Najim. A CLT for Information-theoretic statistics of Gram random matrices with a given variance profile. 2007. ⟨hal-00151136⟩

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