  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.
