ASYMPTOTICS OF SAMPLE EIGENSTRUCTURE FOR A LARGE DIMENSIONAL SPIKED COVARIANCE MODEL

Abstract

This paper deals with a multivariate Gaussian observation model where the eigenvalues of the covariance matrix are all one, except for a finite number which are larger. Of interest is the asymptotic behavior of the eigenvalues of the sample covariance matrix when the sample size and the dimension of the obser- vations both grow to infinity so that their ratio converges to a positive constant. When a population eigenvalue is above a certain threshold and of multiplicity one, the corresponding sample eigenvalue has a Gaussian limiting distribution. There is a phase transition of the sample eigenvectors in the same setting. Another contribution here is a study of the second order asymptotics of sample eigenvectors when corresponding eigenvalues are simple and sufficiently l arge.