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 $n: N=N(n), M=M(n), \gamma^{ -1} \leq \frac{ N}{ n} \leq \gamma$ where $\gamma \in (0,\infty),$ we obtain consistency rates in the form of CLTs for separated spikes tending to infinity fast enough whenever $M$ grows slightly slower than $n: \lim_{ n \to \infty}{ \frac{ \sqrt{ \log{ n}}}{ \log{ \frac{ n}{ M(n)}}}}=0.$ 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.
Categories: math.ST math.PR stat.TH