[FabHK]

I think you're being a bit harsh.

The central limit theorem does hold for iid rv (independent identically distributed random variables) with finite mean and variance. Now, those assumptions can be relaxed (the rv need not be independent, but they can't be too dependent, and they need not have finite variance, but they can't be too "far out"), and some of the pertinent proofs are only a few decades old; but you can hardly expect a survey with 135 proofs to cover all the subtleties.

Some of the other points may be more egregious howlers, but, again, come on - this is not the definite reference for any one of those theorems.

[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.