  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.
