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

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.