  Under the high-dimensional setting that data dimension and sample size tend
to infinity proportionally, we derive the central limit theorem (CLT) for
linear spectral statistics (LSS) of large-dimensional sample covariance matrix.
Different from existing literature, our results do not require the assumption
that the population covariance matrices are bounded. Moreover, many common
kernel functions in the real data such as logarithmic functions and polynomial
functions are allowed in this paper. In our model, the number of spiked
eigenvalues can be fixed or tend to infinity. One salient feature of the
asymptotic mean and covariance in our proposed central limit theorem is that it
is related to the divergence order of the population spectral norm.
