Marchenko Pastur type theorem for independent MRW processes: convergence of the empirical spectral measure

We study the asymptotic of the spectral distribution for large empirical covariance matrices composed of independent Multifractal Random Walk processes. The asymptotic is taken as the observation lag shrinks to 0. In this setting, we show that there exists a limiting spectral distribution whose Stieltjes transform is uniquely characterized by equations which we specify. We also illustrate our results by numerical simulations.