>>12867259we expect normally distributed things to appear in systems that have lots of independent random variables that are roughly independent and roughly the same size (having finite variance that are approximately equal). lots of systems are like this, and that's why you DO see normal distribution all over the place.
systems that are not like this can fail in many ways. if you want to keep things roughly independent, then the kind of processes you find are called self-decomposable distributions. there are interesting ways to describe these distributions, however they are precisely those that appear as limits of the type
where the are independent, but don't have to be identically distributed (each one can be very different), and the are normalizing constants to keep the mean and variance in some bounds.
