Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent random variables with common density function

$$f(x)=\frac{1}{\pi\left(1+x^{2}\right)}$$

Fix $\alpha \in[0,1]$ and set

$$Y_{n}=\operatorname{sgn}\left(X_{n}\right)\left|X_{n}\right|^{\alpha}, \quad S_{n}=Y_{1}+\ldots+Y_{n}$$

Show that for all $\alpha \in[0,1]$ the sequence of random variables $S_{n} / n$ converges in distribution and determine the limit.

[Hint: In the case $\alpha=1$ it may be useful to prove that $\mathbb{E}\left(e^{i u X_{1}}\right)=e^{-|u|}$, for all $\left.u \in \mathbb{R} .\right]$

Show further that for all $\alpha \in[0,1 / 2)$ the sequence of random variables $S_{n} / \sqrt{n}$ converges in distribution and determine the limit.

[You should state clearly any result about random variables from the course to which you appeal. You are not expected to evaluate explicitly the integral

$$m(\alpha)=\int_{0}^{\infty} \frac{x^{\alpha}}{\pi\left(1+x^{2}\right)} d x$$