  In this paper, we establish the central limit theorem (CLT) for linear
spectral statistics (LSSs) of a large-dimensional sample covariance matrix when
the population covariance matrices are involved with diverging spikes. This
constitutes a nontrivial extension of the Bai-Silverstein theorem (BST) (Ann
Probab 32(1):553--605, 2004), a theorem that has strongly influenced the
development of high-dimensional statistics, especially in the applications of
random matrix theory to statistics. Recently, there has been a growing
realization that the assumption of uniform boundedness of the population
covariance matrices in the BST is not satisfied in some fields, such as
economics, where the variances of principal components may diverge as the
dimension tends to infinity. Therefore, in this paper, we aim to eliminate this
obstacle to applications of the BST. Our new CLT accommodates spiked
eigenvalues, which may either be bounded or tend to infinity. A distinguishing
feature of our result is that the variance in the new CLT is related to both
spiked eigenvalues and bulk eigenvalues, with dominance being determined by the
divergence rate of the largest spiked eigenvalues. The new CLT for LSS is then
applied to test the hypothesis that the population covariance matrix is the
identity matrix or a generalized spiked model. The asymptotic distributions of
the corrected likelihood ratio test statistic and the corrected Nagao's trace
test statistic are derived under the alternative hypothesis. Moreover, we
present power comparisons between these two LSSs and Roy's largest root test.
In particular, we demonstrate that except for the case in which the number of
spikes is equal to one, the LSSs could exhibit higher asymptotic power than
Roy's largest root test.
