[tmyklebu]

There's more than enough neat stuff I don't know in there. Unfortunately it's tough to trust a source with so many errors in the stuff I do know.

I wasn't aware of any CLT for iid random variables with infinite variance. Do you have references?

[FabHK]

Check Valentin Petrov Limit Theorems of Probability Theory.

Here [1] is a CLT for RV with infinite variance, Prop 3.1.12, but notice the (larger) scaling coefficient (1/sqrt(n log n)).

Also see the second answer on SO here [2].

[1] https://web.stanford.edu/~montanar/TEACHING/Stat310A/lnotes....

[2] https://stats.stackexchange.com/questions/169611/the-role-of...

EDIT to add: Having said that, the Lindeberg-Feller and the Lyuapunov formulation of the CLT do require finite variance, so maybe I was too quick in stating that that assumption can be relaxed.