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In this paper, we introduce a joint central limit theorem (CLT) for specificbilinear forms, encompassing the resolvent of the sample covariance matrixunder an elliptical distribution. Through an exhaustive exploration of ourtheoretical findings, we unveil a phase transition in the limiting parametersthat relies on the moments of the random radius in our derived CLT.Subsequently, we employ the established CLT to address two statisticalchallenges under elliptical distribution. The first task involves deriving theCLT for eigenvector statistics of the sample covariance matrix. The second taskaims to ascertain the limiting properties of the spiked sample eigenvaluesunder a general spiked model. As a byproduct, we discover that the eigenmatrixof the sample covariance matrix under a light-tailed elliptical distributionsatisfies the necessary conditions for asymptotic Haar, thereby extending theHaar conjecture to broader distributions.

Limiting behavior of bilinear forms for the resolvent of sample covariance matrices under elliptical distribution with applications