A CLT for information-theoretic statistics of non-centered Gram random matrices
Abstract
In this article, we study the fluctuations of a specific linear statistics of the singular values of a large random matrix as the dimensions of the large random matrix go to infinity. This large random matrix is non centered and the random part has a separable variance profile, the random variables either being real or complex. We prove that when centered and properly rescaled, the linear statistics satisfies a Central Limit Theorem and has a Gaussian limit. The variance depends on the second moment of the entries and also on its fourth cumulant. The main motivation comes from the field of wireless communications, where the linear statistics represents the mutual information of a multiple antenna radio channel. This article closely follows the companion article "A CLT for Information-theoretic statistics of Gram random matrices with a given variance profile", Ann. Appl. Probab. (2008) by Hachem et al., however the study of the fluctuations associated to non-centered large random matrices raises specific issues, which are addressed here.
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