  In this paper, we derive a joint central limit theorem for random vector
whose components are function of random sesquilinear forms. This result is a
natural extension of the existing central limit theory on random quadratic
forms. We also provide applications in random matrix theory related to
large-dimensional spiked population models. For the first application, we find
the joint distribution of grouped extreme sample eigenvalues correspond to the
spikes. And for the second application, under the assumption that the
population covariance matrix is diagonal with $k$ (fixed) simple spikes, we
derive the asymptotic joint distribution of the extreme sample eigenvalue and
its corresponding sample eigenvector projection.
