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 observations 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 large.
Key words and phrases: Eigenvalue distribution, principal component analysis, random matrix theory.