The central limit theorems state roughly that under a certain set of properties of a sampling process, the distribution of a statistic from that sample will converge in distribution to the normal distribution.
As the canonical example, let me take the basic central limit theorem: If we take i.i.d. samples $X_1,X_2...$ from a distribution $f$, then the sample average will converfe in diatribution to the normal distribution if
The mean of $f$ exists.
The variance of $f$ exists.
My question is:
Assume that $f$ does not have a variance, and that the sample average of an i.i.d.sample from $f$ does not converge in distribution to the normal distribution.
Are there situations where this sample average nevertheless converges in distribution to some other (non-normal) distribution?
In other words, is there a "central limit theorem" for distributions that don't converge to the nornal distribution?