Relative perturbation bounds with applications to empirical covariance operators

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. Our framework is very general, allowing for a huge variety of stationary, ergodic sequences, requiring only $p > 4$ moments. One of the key ingredients is a specific separation measure for population eigenvalues, which we call the relative rank. Developing a new algebraic approach for relative perturbations, we show that this relative rank gives rise to necessary and sufficient conditions for our concentration inequalities and limit theorems.