[fjkdlsjflkds]

"Central Limit Theorem" says that "adding (bounded variance) stuff together makes it converge to a Gaussian" (roughly). On the other hand, when you are adding random variable together, you are actually convolving their densities. Thus, what the CLT says is that a gaussian is some sort of attractor/fixed point of the "dynamic process" of convolving (finite energy/bounded variance) distributions.

[amitport]

Thanks! Besides bounded variance they should also generally be independent. Or at least I'm not aware of a dependent variable version.

[fjkdlsjflkds]

There are (several) central limit theorems for dependent variables, but you often have to assume other things as well (e.g., stationarity, bounded third moment, and/or limited-range correlations) for it to work.

Example: https://link.springer.com/chapter/10.1007/978-1-4612-0865-5_...