Relative perturbation bounds with applications to empirical covariance operators

TL;DR

This paper develops relative perturbation bounds for empirical covariance operators, providing eigenvalue expansions, concentration inequalities, and limit theorems applicable under broad dependence structures with minimal moment assumptions.

Contribution

It introduces a novel framework for relative perturbation bounds tailored to empirical covariance operators, including a new separation measure called the relative rank.

Findings

Derived eigenvalue and spectral projector expansions

Established concentration inequalities and limit theorems

Applicable to diverse dependence structures with minimal moment conditions

Abstract

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.