Wavelet eigenvalue regression in high dimensions

Abstract

In this paper, we construct the wavelet eigenvalue regression methodology (Abry and Didier (2018bDidier ( , 2018a) 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 non-canonical scaling coordinates and in the presence of additive high-dimensional noise. The measurements are correlated both time-wise and between rows. Building upon the asymptotic and large scale properties of wavelet random matrices in high dimensions, the wavelet eigenvalue regression is shown to be consistent and, under additional assumptions, asymptotically Gaussian in the estimation of the fractal structure of the system. We further construct a consistent estimator of the effective dimension r of the system that significantly increases the robustness of the methodology. The estimation performance over finite samples is studied by means of simulations. * P.A. and H.W. were partially supported by ANR-16-CE33-0020 MultiFracs, France. H.W. was also partially supported by ANR-18-CE45-0007 MUTATION. G.D.'s long term visits to ENS de Lyon were supported by the school, the CNRS and the Simons Foundation collaboration grant #714014. The authors also gratefully acknowledge the support and resources from the Center for High Performance Computing at the University of Utah as well as the high performance computing (HPC) resources and services provided by Technology Services at Tulane University.