ON BILINEAR FORMS BASED ON THE RESOLVENT OF LARGE RANDOM MATRICES

By Walid Hachem, Philippe Loubaton, Jamal Najim, Pascal Vallet. 2010

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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.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b650ca672dc93da47982578591aee9a5346cee50