Divergent spikes and eigenstructure consistency

Several generalizations of Johnstone's spiked model have been considered in recent years, allowing spikes to diverge, which in turn can force the empirical eigenstructure to be consistent with the ground truth. For the simplest such extension, a covariance matrix whose eigenvalues are all one but M of them, the number of features N comparable to the number of samples n with N=N(n), M=M(n), we obtain CLTs for empirical eigenvalues whose deterministic counterparts are separated from the rest and tend to infinity fast enough whenever M grows slightly slower than n. In this talk, our main aim is showing why the most natural CLT in this framework has a mixed centering (a sum of one random and one deterministic term) and how the latter can be converted into a purely random or deterministic quantity, generating thus two new CLTs.