Relative perturbation bounds with applications to empirical covariance operators

Johannes Moritz Jirak, Martin Wahl (Corresponding author)

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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.

Original languageEnglish
Article number108808
Number of pages55
JournalAdvances in Mathematics
Volume412
Early online dateDec 2022
DOIs
Publication statusPublished - 1 Jan 2023

Austrian Fields of Science 2012

  • 101018 Statistics

Keywords

  • Concentration inequalities
  • Covariance operator
  • Limit theorems
  • Perturbation theory
  • Principal component analysis
  • Relative bounds

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