The Central Limit Theorem tells us that for an iid sequence of random variables $(X_n)_{n\geq 0}$ of finite variance $\sigma^2$ and zero mean
$\lim_{n\to\infty}\frac{S_n}{\sqrt{n}}=^d N(0,\sigma^2)$
where $S_n=X_1+\cdots+X_n$.
Suppose we have a similar sequence, except now we suppose that $X_n$ has infinite variance. Then is it possible for the sequence $\frac{S_n}{\sqrt{n}}$ to converge in distribution? Is there always some $\alpha$ such that $n^\alpha S_n$ converges to a non-constant distribution?
(It seems to me that the answer to the first question should be no, but I'm having trouble showing this.)
Thank you.