A result on the almost sure convergence for the R/SR/S statistic

Let {Xi,i≥1}{Xi,i≥1} be a sequence of non-degenerate, i.i.d. random variables, which is in the domain of attraction of the normal law with mean zero and possibly infinite variance, Q(n)=R(n)/S(n)Q(n)=R(n)/S(n) be the rescaled range statistic, where R(n)=max1≤k≤n{∑j=1k(Xj−X̄n)}−min1≤k≤n{∑j=1k(Xj−X̄n)}, S2(n)=∑j=1n(Xj−X̄n)2/n and X̄n=∑j=1nXj/n. Then a result about the almost sure convergence for Q(n)Q(n) is established under a mild condition in this paper.