On bilinear forms based on the resolvent of large random matrices

Details On bilinear forms based on the resolvent of large random matrices On bilinear forms based on the resolvent of large random matrices

2010-04-22

arxiv.org/abs/1004.3848v2

Mathematics

Probability

35 pp. Extended version of the article accepted for publication in Annales de l'Institut Henri Poincar\'e: Probabilit\'e et St

Scientific paper

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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