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

TL;DR

This paper establishes a Marchenko-Pastur type theorem for large covariance matrices formed from independent Multifractal Random Walk processes, describing their spectral distribution as observation lag approaches zero.

Contribution

It introduces a new asymptotic spectral distribution for covariance matrices of independent MRW processes as lag shrinks, with a unique characterization via Stieltjes transform equations.

Findings

Existence of a limiting spectral distribution for MRW-based covariance matrices.

Explicit equations characterizing the Stieltjes transform of the limit.

Numerical simulations confirming theoretical results.

Abstract

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.