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

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.