Title: Limiting Laws for Divergent Spiked Eigenvalues and Largest Non-spiked Eigenvalue of Sample Covariance Matrices,
Abstract: We study the asymptotic distributions of the spiked eigenvalues and the
largest nonspiked eigenvalue of the sample covariance matrix under a general
covariance matrix model with divergent spiked eigenvalues, while the other
eigenvalues are bounded but otherwise arbitrary. The limiting normal
distribution for the spiked sample eigenvalues is established. It has distinct
features that the asymptotic mean relies on not only the population spikes but
also the nonspikes and that the asymptotic variance in general depends on the
population eigenvectors. In addition, the limiting Tracy-Widom law for the
largest nonspiked sample eigenvalue is obtained.
Estimation of the number of spikes and the convergence of the leading
eigenvectors are also considered. The results hold even when the number of the
spikes diverges. As a key technical tool, we develop a Central Limit Theorem
for a type of random quadratic forms where the random vectors and random
matrices involved are dependent. This result can be of independent interest.