A Random Matrix Approximation for the Non-commutative Fractional Brownian Motion

TL;DR

This paper establishes a functional limit theorem for the eigenvalues of matrix fractional Brownian motion, showing convergence to the non-commutative fractional Brownian motion using advanced stochastic calculus tools.

Contribution

It introduces a novel approximation for non-commutative fractional Brownian motion via eigenvalues of matrix fractional Brownian motion, expanding the understanding of non-commutative stochastic processes.

Findings

Eigenvalue empirical measures converge to non-commutative fractional Brownian motion

Utilizes Young and Skorohod integrals and fractional calculus techniques

Provides a new matrix approximation for non-commutative fractional processes

Abstract

A functional limit theorem for the empirical measure-valued process of eigenvalues of a matrix fractional Brownian motion is obtained. It is shown that the limiting measure-valued process is the non-commutative fractional Brownian motion recently introduced by Nourdin and Taqqu. Young and Skorohod stochastic integral techniques and fractional calculus are the main tools used.