If $X_1, X_2, ...$ are i.i.d. Pareto-distributed variables with parameter $\alpha$, I have read (in a heuristic source) that $\frac{1}{n^{1/\alpha}} \sum_{k = 1}^n X_k$ fulfills some law of large numbers, or even a generalized central limit theorem.
Can anyone help me with the exact formulation (i.e. is it the weak or strong law?) or give me a mathematical source where I find this?
Thanks very much in advance!