  In this note, we establish an asymptotic expansion for the centering
parameter appearing in the central limit theorems for linear spectral statistic
of large-dimensional sample covariance matrices when the population has a
spiked covariance structure. As an application, we provide an asymptotic power
function for the corrected likelihood ratio statistic for testing the presence
of spike eigenvalues in the population covariance matrix. This result
generalizes an existing formula from the literature where only one simple spike
exists.
