Optimal eigen expansions and uniform bounds

• 成果类型: Article
• 署名作者: Jirak, Moritz
• 署名单位: Humboldt University of Berlin
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0671-3
• 发表日期: 2016
• 页码: 753-799
• 关键词: functional linear-regression; Principal Component Analysis; time-series; INVARIANCE-PRINCIPLES; ASYMPTOTIC THEORY; dependence; prediction; heteroskedasticity; projections; Operators

摘要

Let be a stationary process with associated lag operators . Uniform asymptotic expansions of the corresponding empirical eigenvalues and eigenfunctions are established under almost optimal conditions on the lag operators in terms of the eigenvalues (spectral gap). In addition, the underlying dependence assumptions are optimal in a certain sense, including both short and long memory processes. This allows us to study the relative maximum deviation of the empirical eigenvalues under very general conditions. Among other things, convergence to an extreme value distribution is shown. We also discuss how the asymptotic expansions transfer to the long-run covariance operator in a general framework.