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In this paper, we characterize the convergence of the (rescaled logarithmic)empirical spectral distribution of wavelet random matrices. We assume amoderately 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 sucha 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 isa random scrambling of a sample of size $n$ of a growing number $p(n)$ offractional processes. Each of the latter processes is a fractional Brownianmotion conditionally on a randomly chosen Hurst exponent. We show that the(rescaled logarithmic) empirical spectral distribution of the wavelet randommatrices converges weakly, in probability, to the distribution of Hurstexponents.

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