Convergence of $\frac{S_{n}}{\sqrt{n\log n}}$

Question

Helo

I need to prove the following statement:

Let $X_{1},X_{2},\ldots,X_{n}$ be $iid$ random variables with density

$$f(x) = \frac{1}{|x|^3}I_{|x|>1}$$

Let $S_{t}=\sum_{i=1}^{n}X_{i}$, show that

$$\frac{S_{n}}{\sqrt{n\log n}}\stackrel{d}{\to}\mathcal{N}(0,1)$$

My take: I have shown that $\mathbb{E}[X]=0$ and that $\mathbb{E}\left[X^{2}\right]$ is not defined, so I cannot use the Central Limit Theorem. Which path shall I take?

Thanks a lot

Answer 1

First, you may show that the characteristic function of $X_1$ is $$ \varphi_X(t)=1-t^2(-\ln|t|+O(1)) \quad \text{as}\quad t\to 0. $$ Then \begin{align} \varphi_{\frac{S_n}{\sqrt{n\ln n}}}(t)&=\left(\varphi_X\!\left(\frac{t}{\sqrt{n\ln n}}\right)\right)^n=\left(1-\frac{t^2}{n\ln n}\left(\ln\frac{\sqrt{n\ln n}}{|t|}+O(1)\right)\right)^n \\ &\to e^{-\frac{1}{2}t^2}\quad \text{as}\quad n\to\infty, \end{align} which is the characteristic function of $N(0,1)$.