  Sample covariance matrix and multivariate $F$-matrix play important roles in
multivariate statistical analysis. The central limit theorems {\sl (CLT)} of
linear spectral statistics associated with these matrices were established in
Bai and Silverstein (2004) and Zheng (2012) which received considerable
attentions and have been applied to solve many large dimensional statistical
problems. However, the sample covariance matrices used in these papers are not
centralized and there exist some questions about CLT's defined by the
centralized sample covariance matrices. In this note, we shall provide some
short complements on the CLT's in Bai and Silverstein (2004) and Zheng (2012),
and show that the results in these two papers remain valid for the centralized
sample covariance matrices, provided that the ratios of dimension $p$ to sample
sizes $(n,n_1,n_2)$ are redefined as $p/(n-1)$ and $p/(n_i-1)$, $i=1,2$,
respectively.
