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Let \{$X_{ij}$\}, $i,j=...,$ be a double array of i.i.d. complex randomvariables with $EX_{11}=0,E|X_{11}|^2=1$ and $E|X_{11}|^4<\infty$, and let$A_n=\frac{1}{N}T_n^{{1}/{2}}X_nX_n^*T_n^{{1}/{2}}$, where $T_n^{{1}/{2}}$ isthe square root of a nonnegative definite matrix $T_n$ and $X_n$ is the$n\times N$ matrix of the upper-left corner of the double array. The matrix$A_n$ can be considered as a sample covariance matrix of an i.i.d. sample froma population with mean zero and covariance matrix $T_n$, or as a multivariate$F$ matrix if $T_n$ is the inverse of another sample covariance matrix. Toinvestigate the limiting behavior of the eigenvectors of $A_n$, a new form ofempirical spectral distribution is defined with weights defined by eigenvectorsand it is then shown that this has the same limiting spectral distribution asthe empirical spectral distribution defined by equal weights. Moreover, if\{$X_{ij}$\} and $T_n$ are either real or complex and some additional momentassumptions are made then linear spectral statistics defined by theeigenvectors of $A_n$ are proved to have Gaussian limits, which suggests thatthe eigenvector matrix of $A_n$ is nearly Haar distributed when $T_n$ is amultiple of the identity matrix, an easy consequence for a Wishart matrix.

On asymptotics of eigenvectors of large sample covariance matrix