  In this paper, we mathematically construct wavelet eigenanalysis in high
dimensions (Abry and Didier (2018a, 2018b)) by characterizing the scaling
behavior of the eigenvalues of large wavelet random matrices. We assume that
possibly non-Gaussian, finite-variance $p$-variate measurements are made of a
low-dimensional $r$-variate ($r \ll p$) fractional stochastic process with
non-canonical scaling coordinates and in the presence of additive
high-dimensional noise. We show that the $r$ largest eigenvalues of the wavelet
random matrices, when appropriately rescaled, converge to scale invariant
functions in the high-dimensional limit. By contrast, the remaining $p-r$
eigenvalues remain bounded. In addition, we show that, up to a log
transformation, the $r$ largest eigenvalues of wavelet random matrices exhibit
asymptotically Gaussian distributions. We further show how the asymptotic and
large-scale behavior of wavelet eigenvalues can be used to construct
statistical inference methodology for a high-dimensional signal-plus-noise
system.
