On high-dimensional wavelet eigenanalysis

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.