Asymptotics of Eigenvalues and Unit-Length Eigenvectors of Sample Variance and Correlation Matrices

Multivariate asymptotic (normal) distributions for eigenvalues and unit-length eigenvectors of sample variance and correlation matrices are derived. Beside the general case, when existence of the (finite) fourth-order moments of the population distribution is assumed, formulae for the asymptotic variance matrices in the cases of normal and elliptical populations are also derived. It is assumed throughout that population variance and correlation matrices are nonsingular and without multiple eigenvalues.