Let {$X_i$, $i{geq}1$} be a sequence of i.i.d. nondegenerate random variables which is in the domain of attraction of the normal law with mean zero and possibly infinite variance. Denote $S_n={sum}_{i=1}^n;X_i$, $M_n=max_{1{leq}i{leq}n};{mid}S_i{mid}$ and $V_n^2={sum}_{i=1}^n;X_i^2$. Then for d > -1, we showed that under some regularity conditions, $$lim_{{varepsilon}{searrow}0}{varepsilon}^2^{d+1}sum_{n=1}^{infty}frac{(loglogn)^d}{nlogn}I{M_n/V_n{geq}sqrt{2loglogn}({varepsilon}+{alpha}_n)}=frac{2}{sqrt{pi}(1+d)}{Gamma}(d+3/2)sum_{k=0}^{infty}frac{(-1)^k}{(2k+1)^{2d+2}};a.s.$$ holds in this paper, where If g denotes the indicator function.