On the empirical spectral distribution of large wavelet random matrices based on mixed-Gaussian fractional measurements in moderately high dimensions

Abstract

In this paper, we characterize the convergence of the (rescaled logarithmic) empirical spectral distribution of wavelet random matrices. We assume a moderately high-dimensional framework where the sample size $n$, the dimension $p(n)$ and, for a fixed integer $j$, the scale $a(n)2^j$ go to infinity in such a way that $\lim_{n \rightarrow \infty}p(n)\cdot a(n)/n = \lim_{n \rightarrow \infty} o(\sqrt{a(n)/n})= 0$. We suppose the underlying measurement process is a random scrambling of a sample of size $n$ of a growing number $p(n)$ of fractional processes. Each of the latter processes is a fractional Brownian motion conditionally on a randomly chosen Hurst exponent. We show that the (rescaled logarithmic) empirical spectral distribution of the wavelet random matrices converges weakly, in probability, to the distribution of Hurst exponents.

Figures (2)

• Figure 1: The distribution of the (rescaled logarithmic) wavelet e.s.d. in the three-way limit \ref{['e:three-fold_lim']}. A Monte Carlo study displays a tri-modal distribution emerging in the rescaled logarithmic wavelet e.s.d. in the three-way limit \ref{['e:three-fold_lim']} (n.b.: after applying an affine transformation, the results are shown on the same scale as that of the distribution $\pi(dH)$). In the depicted simulation study based on $1000$ realizations, $\pi(dH)$ is a discrete uniform distribution with support $\{0.2,0.5,0.8\}$. From left to right, respectively, $(\textnormal{sample size}, \textnormal{scale},\textnormal{dimension} ) = (2^{10}, 2^4, 2^3),(2^{15}, 2^5, 2^5)$ and $(2^{18}, 2^6, 2^6)$. Wavelet log-eigenvalues weighted over multiple scales were used for enhanced ("debiased") finite-sample convergence (cf. Abry and Didier abry:didier:2018:n-variate and Abry et al. abry:boniece:didier:wendt:2022).
• Figure 2: Schematic representation of the "forces" acting on the rescaled log-eigenvalues in the three-way limit \ref{['e:three-fold_lim']}. We can interpret that two distinct "fractal forces" -- one repulsive, one attractive -- act on the rescaled wavelet log-eigenvalues. In the three-way limit \ref{['e:p(n),a(n)_conditions']}, as a result of these forces only the large-scale features of the system remain. We are then left with the distribution of the Hurst exponents, as illustrated in Figure (see Remark for a discussion).