A CLT FOR INFORMATION-THEORETIC STATISTICS OF GRAM RANDOM MATRICES WITH A GIVEN VARIANCE PROFILE

By Walid Hachem, Philippe Loubaton, Jamal Najim. 2007

93 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9ca7bda6b27efd807bfcb6e260e2f54b763a825f

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Citations in the corpus

Large System Analysis of Linear Precoding in Correlated MISO Broadcast Channels Under Limited Feedback (2009, methodology): “…of B N , used in random matrix theory, e.g. by Mar˘cenko and Pastur [25] to derive the limiting distribution of sample covariance matrices (which corresponds here to the case Remark 1: If X N is Gaussian, besides (3), the authors infer that, by applying the Gaussian method [26], it can be…”
Fluctuations of eigenvalues and second order Poincaré inequalities (2007, background): “[28, 29] provide several illuminating insights and new results.”
Random matrix theory in statistics: A review (2014, background): “, 2012), and for data matrices having independent entries with a given variance profile (Hachem et al., 2008).”