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),γ−1≤Nn≤γ where γ∈(0,∞), we obtain consistency rates in the form of CLTs for separated spikes tending to infinity fast enough whenever M grows slightly slower than n:limn→∞ logn logn 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(n1∕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.