Eigenvectors of some large sample covariance matrix ensembles

We consider sample covariance matrices ${S_N=\frac{1}{p}\Sigma_N^{1/2}X_NX_N^* \Sigma_N^{1/2}}$ where X _N is a N ×? p real or complex matrix with i.i.d. entries with finite 12th moment and ?? _N is a N ×? N positive definite matrix. In addition we assume that the spectral measure of ?? _N almost surely converges to some limiting probability distribution as N ?? ?? and p/N ?? ?? >?0. We quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type ${\frac{1}{N}\text{Tr} ( g(\Sigma_N) (S_N-zI)^{-1}),}$ where I is the identity matrix, g is a bounded function and z is a complex number. This is then used to compute the asymptotically optimal bias correction for sample eigenvalues, paving the way for a new generation of improved estimators of the covariance matrix and its inverse.