On highdimensional wavelet eigenanalysis
Abstract
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 nonGaussian, finitevariance $p$variate measurements are made of a lowdimensional $r$variate ($r \ll p$) fractional stochastic process with noncanonical scaling coordinates and in the presence of additive highdimensional 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 to scale invariant functions in the highdimensional limit. By contrast, the remaining $pr$ eigenvalues remain bounded. Under additional assumptions, we show that, up to a log transformation, the $r$ largest eigenvalues of wavelet random matrices exhibit asymptotically Gaussian distributions. The results have direct consequences for statistical inference.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 arXiv:
 arXiv:2102.05761
 Bibcode:
 2021arXiv210205761A
 Keywords:

 Mathematics  Statistics Theory;
 Mathematics  Probability;
 60G22;
 60B20
 EPrint:
 56 pages, 1 figure. Major revision. Main results now established under more general conditions