Local independence of fractional Brownian motion

TL;DR

This paper proves that the sigma-algebras generated by differences of fractional Brownian motion over small, disjoint intervals become asymptotically independent as the interval size shrinks to zero, using mutual information.

Contribution

It establishes the asymptotic independence of local sigma-algebras for fractional Brownian motion with quantitative estimates and generalizations.

Findings

Mutual information between local sigma-algebras tends to zero as interval size shrinks.

Asymptotic independence holds for fractional Brownian motion with Hurst index H.

Provides quantitative bounds on the rate of mutual information decay.

Abstract

Let S(t,t') be the sigma-algebra generated by the differences X(s)-X(s) with s,s' in the interval(t,t'), where (X_t) is the fractional Brownian motion process with Hurst index H between 0 and 1. We prove that for any two distinct t and t' the sigma-algebras S(t-a,t+a) and S(t'-a,t'+a) are asymptotically independent as a tends to 0. We show this in the strong sense that Shannon's mutual information between these two sigma-algebras tends to zero as a tends to 0. Some generalizations and quantitative estimates are provided also.