Precise asymptotics in the law of the iterated logarithm for R/S statistic

Let {X,Xn,n≥1} be a sequence of i.i.d. random variables which is in the domain of attraction of the normal law with zero mean and possibly infinite variance, Q(n)=R(n)/S(n) be the rescaled range statistic, where R(n)=max1≤k≤n{∑j=1k(Xj−X¯n)}−min1≤k≤n{∑j=1k(Xj−X¯n)}, S2(n)=∑j=1n(Xj−X¯n)2/n and X¯n=∑j=1nXj/n. Then two precise asymptotics related to probability convergence for Q(n) statistic are established under some mild conditions in this paper. Moreover, the precise asymptotics related to almost surely convergence for Q(n) statistic is also considered under some mild conditions.