TITLE: CLT for random variables with heavy tails
QUESTION [5 upvotes]: The simplest form of classic CLT can be put in the following form:
If $X_1,X_2,\dots$ are i.i.d. with mean $0$ and variance $1$, then the distribution of the normalized sum $$\frac{X_1+X_2+\dots+X_n}{\sqrt{n}}$$ converges to that of the Gaussian.
Is there any result as following?
If $X_i$ are i.i.d. with infinite variance, then $$\frac{X_1+X_2+\dots+X_n}{n^{1/\alpha}}$$ converges in distribution to a Cauchy random variable. 
If so, what should be $\alpha$?

REPLY [5 votes]: There are results of this form, but the sample mean doesn't necessarily converge to a Cauchy distribution--there's a whole family of distributions called the stable distributions which can be limits like this. The Normal and Cauchy distributions are just two of them, and all but the Normal have infinite variance, so just having infinite variance isn't enough to converge to the Cauchy--you need exactly the right tail behavior.
In addition to there being infinitely many possible limiting distributions (including asymmetric ones), there are infinitely many possible values of $\alpha$ in your partial sum. In the case where the $X_i$ are i.i.d. Cauchy, we can take $\alpha = 1$ and the mean we get is also Cauchy. In other cases, though, if the distributions have different tail behavior and converge to a different stable distribution, normalizing factor will be different.