Abstract : Let $\{X_i, i\ge 1\}$ 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 $S_n=\sum_{i=1}^n X_i, M_n=\max_{1\le i\le n}|S_i|$ and $V_n^2=\sum_{i=1}^n X_i^2$. Then for $d>-1$, we showed that under some regularity conditions, \begin{align*} \lim_{\varepsilon\searrow0}\varepsilon^{2(d+1)}\sum_{n=1}^{\infty} \frac{(\log\log n)^{d}}{n\log n}I\{M_{n}/V_n \geq \sqrt{2\log\log n}(\varepsilon+\alpha_{n})\} = \frac{2}{\sqrt{\pi}(1+d)}\Gamma(d+3/2)\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(2k+1)^{2d+2}}~~ {\rm a.s.} \end{align*} holds in this paper, where $I\{\cdot\}$ denotes the indicator function.

Keywords : almost sure convergence, self-normalized, domain of attraction of the normal law, law of the iterated logarithm