Some correlations are fine though, there are versions of CLT that applies even when there are benign correlations.

https://en.wikipedia.org/wiki/Central_limit_theorem#Dependen...

I know you know that and were just simplifying. Just wanted this fact to be better known for practitioners. Your comment on multiplicative processes is spot on.

I say more here

https://news.ycombinator.com/item?id=47437152

It's bit of a shame that these other limiting distributions are not as tractable as the Gaussian.