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On high-dimensional wavelet eigenanalysis
Journal article   Open access   Peer reviewed

On high-dimensional wavelet eigenanalysis

Patrice Abry, B. Cooper Boniece, Gustavo Didier and Herwig Wendt
The Annals of applied probability, v 34(6)
01 Dec 2024
DOI: https://doi.org/10.1214/24-AAP2092
url
https://arxiv.org/abs/2102.05761View
Submitted Open arXiv.org - Non-exclusive license to distribute

Abstract

Wavelets operator self-similarity random matrices
<p>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.</p>

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Statistics & Probability
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