Consider sample covariance matrices of the form $Q:=\Sigma^{1/2} X X^*
\Sigma^{1/2}$, where $X=(x_{ij})$ is an $n\times N$ random matrix whose entries
are independent random variables with mean zero and variance $N^{-1}$, and
$\Sigma$ is a deterministic positive-definite matrix. We study the limiting
behavior of the eigenvectors of $Q$ through the so-called eigenvector empirical
spectral distribution (VESD) $F_{\mathbf u}$, which is an alternate form of
empirical spectral distribution with weights given by $|\mathbf u^\top
\xi_k|^2$, where $\mathbf u$ is any deterministic unit vector and $\xi_k$ are
the eigenvectors of $Q$. We prove a functional central limit theorem for the
linear spectral statistics of $F_{\mathbf u}$, indexed by functions with
H{\"o}lder continuous derivatives. We show that the linear spectral statistics
converge to universal Gaussian processes both on global scales of order 1, and
on local scales that are much smaller than 1 and much larger than the typical
eigenvalues spacing $N^{-1}$. Moreover, we give explicit expressions for the
means and covariance functions of the Gaussian processes, where the exact
dependence on $\Sigma$ and $\mathbf u$ allows for more flexibility in the
applications of VESD in statistical estimations of sample covariance matrices.Comment: 60 pages, 2 figure