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 t 1 and t 2 the sigma-algebras σ ( t 1 − ε , t 1 + ε ) and σ ( t 2 − ε , t 2 + 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.