On the eigenstructure of covariance matrices with divergent spikes

Abstract

For a generalization of Johnstone’s spiked model, a covariance matrix with eigenvalues all one but M of them, the number of features N comparable to the number of samples $n:N=N(n),M=M(n),{\mathit{\gamma }}^{-1}\le \frac{N}{n}\le \mathit{\gamma }$ where $\mathit{\gamma }\in (0,\mathrm{\infty })$, we obtain consistency rates in the form of CLTs for separated spikes tending to infinity fast enough whenever M grows slightly slower than $n:{lim}_{n\to \mathrm{\infty }}\frac{\sqrt{logn}}{log\frac{n}{M(n)}}=0$. Our results fill a gap in the existing literature in which the largest range covered for the number of spikes has been $o(n^{1∕6})$ and reveal a certain degree of flexibility for the centering in these CLTs inasmuch as it can be empirical, deterministic, or a sum of both. Furthermore, we derive consistency rates of their corresponding empirical eigenvectors to their true counterparts, which turn out to depend on the relative growth of these eigenvalues.