On high-dimensional 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 non-Gaussian, finite-variance p-variate measurements are made of a low-dimensional r-variate ($\mathit{r}\ll \mathit{p}$) 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 $\mathit{p}-\mathit{r}$ 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.