UCLA

In this work we study the extreme eigenvalues and eigenvectors of sample correlation matrices arising from Johnstone's spiked model, or, a factor model. We make the important assumption that the model has weak factors. Under the assumption of 6 bounded moments and the assumption that the eigenvalues are comparable and not too close to one another, we show that the distribution of the spiked eigenvalues of the sample correlation matrix are close enough to those of the sample covariance matrix to have the same distribution. We show a similar result for the eigenvectors under the additional assumption that the eigenvalues are bounded. We also show that the non-spiked eigenvalues of the sample correlation matrix are close to those of an appropriate random sample covariance matrix, which allows us to establish universal Tracy-Widom statistics if the factors are weak enough. However, we establish a phase transition, whereby the non-spiked eigenvalues may have asymptotically Gaussian distribution if the factors are not weak enough.