Convergence of the spectrum of empirical covariance matrices for independent MRW processes

¹ CNRS, UMR 7534, 75016 Paris, France.
² Université Paris-Dauphine, Ceremade, 75016 Paris, France.
allez@ceremade.dauphine.fr; rhodes@ceremade.dauphine.fr; vargas@ceremade.dauphine.fr

Received: 3 December 2013
Revised: 12 September 2014

Abstract

We study the asymptotic of the spectral distribution for large empirical covariance matrices composed of independent lognormal 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.