A CLT for the LSS of large dimensional sample covariance matrices with diverging spikes

In this paper, we establish the central limit theorem (CLT) for linear spectral statistics (LSS) of large-dimensional sample covariance matrix when the population covariance matrices are not uniformly bounded, which is a nontrivial extension of the Bai-Silverstein theorem (BST) (2004). The latter has strongly influenced the development of high-dimensional statistics, especially in applications of random matrix theory to statistics. However, the assumption of uniform boundedness of the population covariance matrices has seriously limited the applications of the BST. The aim of this paper is to remove the barriers for the applications of the BST. The new CLT, allows spiked eigenvalues to exist, which may be bounded or tend to infinity. An important feature of our result is that the roles of either spiked eigenvalues or the bulk eigenvalues predominate in the CLT, depending on which variance is nonnegligible in the summation of the variances. The CLT for LSS is then applied to compare four linear hypothesis tests: The Wilk's likelihood ratio test, the Lawly-Hotelling trace test, the Bartlett-Nanda-Pillai trace test, and Roy's largest root test. We also derive and analyze their power function under particular alternatives.