Local independence of fractional Brownian motion

Let σ_(t,t′) be the sigma-algebra generated by the differences X_s−X_s′ with s,s′∈(t,t′), where (X_t)_−∞<t<∞ is the fractional Brownian motion with Hurst index H∈(0,1). We prove that for any two distinct timepoints t₁ and t₂ the sigma-algebras σ_{(t1−ε,t1+ε)} and σ_{(t2−ε,t2+ε)} are asymptotically independent as ε↘0. We show the independence in the strong sense that Shannon’s mutual information between the two σ-algebras tends to zero as ε↘0. Some generalizations and quantitative estimates are also provided.