Let \(\{Y_i: -\infty(1+\epsilon) \sqrt{2n \log \Psi(n)}\Bigg\}\begin{cases} 0, \\ =\infty &\text{ if }\epsilon <0,\end{cases} \] if and only if \(\operatorname{E}Y_1=0 \text{ and } \operatorname{E}Y_1^2=1\). A couple of corollaries are also obtained when \(h(n)\) takes special forms such as (i) \(h(n)=(\log\log n)^{b}\), \(b\geq 0\), (ii) \(h(n)=(\log n)^{r}\), \(0\leq r<1\), and \(h(n)=\log n\).