ON HIGH-DIMENSIONAL WAVELET EIGENANALYSIS

• 成果类型: Article
• 署名作者: Abry, Patrice; Boniece, B. cooper; Didier, Gustavo; Wendt, Herwig
• 署名单位: Universite Paris Cite; Ecole Normale Superieure de Lyon (ENS de LYON); Centre National de la Recherche Scientifique (CNRS); Drexel University; Tulane University; Universite Federale Toulouse Midi-Pyrenees (ComUE); Universite de Toulouse; Institut National Polytechnique de Toulouse; Universite Toulouse III - Paul Sabatier; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2092
• 发表日期: 2024
• 页码: 5287-5350
• 关键词: sample covariance matrices; empirical spectral distribution; memory parameter; long-memory; time-series; LARGEST EIGENVALUE; cointegration; UNIVERSALITY; regression; estimator

摘要

In this paper, we characterize the asymptotic and large scale behavior of the eigenvalues of wavelet random matrices in high dimensions. We assume that possibly non-Gaussian, finite-variance p-variate measurements are made of a low-dimensional r-variate (r << p ) fractional stochastic process with noncanonical scaling coordinates and in the presence of additive high- dimensional noise. The measurements are correlated both timewise and between rows. We show that the r largest eigenvalues of the wavelet random matrices, when appropriately rescaled, converge in probability to scale-invariant functions in the high-dimensional limit. By contrast, the remaining p - r eigenvalues remain bounded in probability. Under additional assumptions, we show that the r largest log-eigenvalues of wavelet random matrices exhibit asymptotically Gaussian distributions. The results have direct consequences for statistical inference.