  The goal of this paper is to establish relative perturbation bounds, tailored
for empirical covariance operators. Our main results are expansions for
empirical eigenvalues and spectral projectors, leading to concentration
inequalities and limit theorems. One of the key ingredients is a specific
separation measure for population eigenvalues, which we call the relative rank,
giving rise to a sharp invariance principle in terms of limit theorems,
concentration inequalities and inconsistency results. Our framework is very
general, requiring only $p > 4$ moments and allows for a huge variety of
dependence structures.
