  In this paper, we introduce a joint central limit theorem (CLT) for specific
bilinear forms, encompassing the resolvent of the sample covariance matrix
under an elliptical distribution. Through an exhaustive exploration of our
theoretical findings, we unveil a phase transition in the limiting parameters
that relies on the moments of the random radius in our derived CLT.
Subsequently, we employ the established CLT to address two statistical
challenges under elliptical distribution. The first task involves deriving the
CLT for eigenvector statistics of the sample covariance matrix. The second task
aims to ascertain the limiting properties of the spiked sample eigenvalues
under a general spiked model. As a byproduct, we discover that the eigenmatrix
of the sample covariance matrix under a light-tailed elliptical distribution
satisfies the necessary conditions for asymptotic Haar, thereby extending the
Haar conjecture to broader distributions.
