Central Limit Theorem

The Central Limit Theorem is about convolutions

If you have a bunch of distributions $f\_i$ (say, $n$ of them), and you convolve them all together into a distribution $F^∗:=f_1∗f_2∗f_3...∗f_n$, then the larger $n$ is, the more $F^∗$  will resemble a Gaussian distribution.

 

The simplest version of the central limit theorem requires that the distributions $f_i$ must be

1. Independent, and
2. Identically distributed.

Note that the density function of the sum of two random variables is the convolution of two densitiy functions.