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

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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Contributor : Walid Hachem Connect in order to contact the contributor
Submitted on : Friday, June 1, 2007 - 4:35:53 PM
Last modification on : Saturday, January 15, 2022 - 3:58:48 AM
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


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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