Let $X_1, X_2,...$ be iid with density $f(x)={\begin{cases} 0, if \mid x \mid \leq 1 \\ \mid x \mid^{-3} , if\mid x \mid >1\end{cases}}$. Prove that $(n\log{n})^{-1/2}\Sigma_{i=1}^{n} X_i \rightarrow \mathcal N(0,\sigma^2)$ in distribution.
My thoughts: I want to usc CLT on this problem and I can do some transformations: $original=\frac{\Sigma_{i=1}^{n} X_i - 0}{\sigma \sqrt n} \times \frac{\sigma}{\sqrt{\log n}}$ and I'm stuck here. The hint is to truncate $X_i$ at $\pm \sqrt{n\log n}$, but I'm a little confused by this hint.