  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 stimulated the development of high-dimensional statistics,
especially the application of random matrix theory to statistics. However, the
assumption of uniform boundedness of the population covariance matrices is
found strongly limited to the applications of BST. The aim of this paper is to
remove the blockages to the applications of BST. The new CLT, allows the spiked
eigenvalues to exist and tend to infinity. It is interesting to note that the
roles of either spiked eigenvalues or the bulk eigenvalues or both of the two
are dominating in the CLT.
  Moreover, the results are checked by simulation studies with various
population settings. The CLT for LSS is then applied for testing the hypothesis
that a covariance matrix $ \bSi $ is equal to an identity matrix. For this, the
asymptotic distributions for the corrected likelihood ratio test (LRT) and
Nagao's trace test (NT) under alternative are derived, and we also propose the
asymptotic power of LRT and NT under certain alternatives.
