Local independence of fractional Brownian motion

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.