Abstract: Let {X(t), t > 0} be a fractional Brownian motion of order 2α with 0 < α <1,β > 0 be a real number, a_T be a function of T and 0 < a_T ≤ T, lim_T→∞(log T/a_T)/log log T=r, (0 ≤ r ≤ ∞). In this paper, we proved that
c₁((r)/1+r)^α ≤ lim inf_T→∞(log log T)^β max_{aT ≤ t ≤ T}(|X(T)+X(T-t)|)/t^α(log(T/t)+log logt)^β ≤ c²((r)/1+r)^α,a.s. where c₁, c₂ are two positive constants depending only on α,β.

Key words: Hanson-Russo type increments, Wiener process, fractional Brownian motion