On the Eigenstructure of Covariance Matrices with Divergent Spikes
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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≤N/n≤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: lim_n →∞√(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(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.
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