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Consider a matrix $\Sigma_n$ with random independent entries, eachnon-centered with a separable variance profile. In this article, we study thelimiting 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 atthe 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-centeredGram matrices such as central limit theorems, individual entries of theresolvent, and eigenvalue separation.

On bilinear forms based on the resolvent of large random matrices