Quantitative limit theorems and bootstrap approximations for empirical spectral projectors

• Published in: Probability theory and related fields Vol. 190; no. 1-2; pp. 119 - 177
• Main Authors: Jirak, Moritz, Wahl, Martin
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2024; Springer Nature B.V
• Subjects: Approximation; Covariance; Economics; Eigenvalues; Finance; Hilbert space; Inequality; Insurance; Management; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Probability Theory and Stochastic Processes; Projectors; Quantitative Finance; Random variables; Statistics for Business; Theorems; Theoretical; Normal approximation; Relative perturbation theory; Covariance operator; 62G09; 47A55; Principal components analysis; Bootstrap; Relative rank; 60F05; Spectral projector; 62H25
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-024-01290-4

Given finite i.i.d. samples in a Hilbert space with zero mean and trace-class covariance operator Σ , the problem of recovering the spectral projectors of Σ naturally arises in many applications. In this paper, we consider the problem of finding distributional approximations of the spectral projectors of the empirical covariance operator Σ ^ , and offer a dimension-free framework where the complexity is characterized by the so-called relative rank of Σ . In this setting, novel quantitative limit theorems and bootstrap approximations are presented subject to mild conditions in terms of moments and spectral decay. In many cases, these even improve upon existing results in a Gaussian setting.