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.Comment: Comparing with the old manuscript, we modified the title of the
paper. arXiv admin note: text overlap with arXiv:2205.07280. arXiv admin
note: text overlap with arXiv:2205.0728