Central Limit Theorems for the Unbounded Spiked Eigenvalues and the largest non-spike of Sample Covariance Matrices

Abstract

Consider a spiked population covariance matrix with the first K largest eigenvalues tending to infinity (spikes) and the remaining eigenvalues being bounded. We establish the asymptotic joint distributions for the spiked eigenvalues of its sample covariance matrices when the population spikes satisfy some conditions. The number of the spikes K is allowed to diverge with a certain rate. Furthermore, the largest non-spiked eigenvalue is also shown to converge in distribution to Type-1 Tracy-Widom distribution under some mild conditions.