On high-dimensional wavelet eigenanalysis

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 (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. 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. 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.

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