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 where , we obtain consistency rates in the form of CLTs for separated spikes tending to infinity fast enough whenever M grows slightly slower than . Our results fill a gap in the existing literature in which the largest range covered for the number of spikes has been 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.
Acknowledgments
The author would like to thank professors George Papanicolaou and Lenya Ryzhik for their comments and suggestions, especially for the feedback concerning the expository aspects of this paper, and the referee, whose remarks helped winnow out several typos.
Citation
Simona Diaconu. "On the eigenstructure of covariance matrices with divergent spikes." Bernoulli 29 (2) 1275 - 1296, May 2023. https://doi.org/10.3150/22-BEJ1498