On bilinear forms based on the resolvent of large random matrices

Abstract : 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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Submitted on : Tuesday, August 23, 2011 - 1:25:06 PM
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Walid Hachem, Philippe Loubaton, Jamal Najim, Pascal Vallet. On bilinear forms based on the resolvent of large random matrices. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institute Henri Poincaré, 2013, 49 (1), pp.36-63. ⟨10.1214/11-AIHP450⟩. ⟨hal-00474126v2⟩

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