[YouWhy]

A minor thought I found inspiring many years ago: the Central Limit Theorem may be considered a statement about dynamic processes in the functional space of distributions.

If you consider the operator of convolution (rescaling for unit variance), the normal distribution is the only attractor (under many "natural" choices of metrics).

[Sniffnoy]

It's actually not the only attractor -- it's just the only attractor once you restrict to finite variance. If you allow infinite variance, there can be others: https://en.wikipedia.org/wiki/Stable_distribution

[jesuslop]

Makes sense, if you have a random process with IID variables, the distributions d are the same, so you can view the partial sums process (∑Xi) dynamic as advancing by convolving with d.

In the generating function pic, the distribution is borne as coefficients of polynomials or of formal power series, and convolution is polynomial multiplication. One example of your step is multiplying by (x+1)/2 (Bernoulli trial), and that gives approaches to the gaussian by chunks of normalized binomial coefficients.

Other related limit, described as a functional central limit theorem is given by Donsker's theorem [1], giving the passage from the discrete situation (random walk) to the continous (Brownian motion).

[1] https://en.wikipedia.org/wiki/Donsker%27s_theorem

[amitport]

Can you elaborate / add reference?

[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_...