We are asked to prove that given $\{X_n\}$ being a sequence of iid r.v's with density $|x|^{-3}$ outside $(-1,1)$, the following is true: $$ \frac{X_1+X_2 + \dots +X_n}{\sqrt{n\log n}} \xrightarrow{\mathcal{D}}N(0,1). $$
My idea is to use the taylor expansion of the characteristic function. But no matter what I do, I run into trouble with infinity and I cannot prove the convergence of the limit of c.f.
Can anybody give a hint? Thanks so much!
This question was asked an other time on Crosss Validated. One of the ideas of proof is to use Lindeberg central limit theorem to the array of random variables $$ Y_{n,k}:=X_k\mathbf 1\left\{\left\lvert X_k\right\rvert\leqslant n\right\} $$ and show that the contribution of the partial sums of $X_k\mathbf 1\left\{\left\lvert X_k\right\rvert\gt n\right\} $ is negligible.