Abstract
Consider an N×n random matrix Yn=(Ynij) with entries given by
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/n→c∈(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.
Citation
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
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