Asymptotics of the leading sample eigenvalues for a spiked covariance model, Available at http://www-stat.stanford.edu/ debashis

Abstract: We consider a multivariate Gaussian observation model where the covariance matrix is diagonal and the diagonal entries are all equal to one except for a finite number which are bigger. We address the question of asymptotic behaviour of the eigenvalues of the sample covariance matrix when the sample size and the dimension of the observations both grow to infinity in such a way that their ratio converges to a positive constant. We establish almost sure limits of the largest few sample eigenvalues. We also show that when a population eigenvalue is above a certain threshold and of multiplicity one, the corresponding sample eigenvalue has a Gaussian limiting distribution. We also demonstrate a phase transition phenomenon of the sample eigenvectors in the same setting