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Pré-Publication, Document De Travail Année : 2010

On bilinear forms based on the resolvent of large random matrices

Résumé

Consider a matrix $\Sigma_n$ with random independent entries, each non-centered with a separable variance profile. In this article, we study the limiting behavior of the random bilinear form $u_n^* Q_n(z) v_n$, where $u_n$ and $v_n$ are deterministic vectors, and Q_n(z) is the resolvent associated to $\Sigma_n \Sigma_n^*$ as the dimensions of matrix $\Sigma_n$ go to infinity at the same pace. Such quantities arise in the study of functionals of $\Sigma_n \Sigma_n^*$ which do not only depend on the eigenvalues of $\Sigma_n \Sigma_n^*$, and are pivotal in the study of problems related to non-centered Gram matrices such as central limit theorems, individual entries of the resolvent, and eigenvalue separation.
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Dates et versions

hal-00474126 , version 1 (19-04-2010)
hal-00474126 , version 2 (23-08-2011)

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Walid Hachem, Philippe Loubaton, Jamal Najim, Pascal Vallet. On bilinear forms based on the resolvent of large random matrices. 2010. ⟨hal-00474126v1⟩
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