Consider the following exam quesiton
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 $u \in \mathbb{R}$.]
I was able to do the case for $\alpha=1$, in particular I used contour integration to find the characteristic function and then used Levy's Continuity Theorem, and then the fact that weak convergence is equivalent to convergence in distribution. Using that I got that $S_n /n$ converges to a Cauchy variable (in distribution).
I tried to do the same with $\alpha \neq1$, however, doing the integral proved challenging as when I tried to elevate in the complex plane.I introduced a branch point.
How does one attack the $\alpha \neq 1$ case?