Understanding Gaussians

Great article, but I wish it would have made a more explicit mention of the* central limit theorem (CLT), which I think is what makes the normal distribution "normal." For those not familiar, here is the jist: suppose you have `n` independent, finite-variance random variables with support in the real numbers (so things like count R.V.s work). Asymptotically, as n->infinity, the distribution of the mean will approach a normal distribution. Usually, n doesn't have to be big for this to be a reasonable approximation. n~30 is often fine. The CLT extends in a

To me, this is one of the most astonishing things about probability theory, as well as one of the most useful.

The normal distribution is just one of a class of "stable distributions," all sharing the properties of sums of their R.V.s being in the same family.

The same idea can be generalized much further. The underlying idea is the distribution of "things" as they get asymptotically "bigger." The density of eigenvalues of random matrices with I.I.D entries approach the Wigner Semicircle Distribution, which is exactly what it sounds like. It plays the role of the normal distribution in the very practically-promising theory of free (noncommutative) probability.

https://en.wikipedia.org/wiki/Wigner_semicircle_distribution

Further reading:

https://terrytao.wordpress.com/2010/01/05/254a-notes-2-the-c...

*there's a few normal distribution CLTs, but this is the intuitive one that usually matters in practice

Good for you for stating the assumptions properly that go into the CLT and for mentioning other stable distributions.

I disagree about the Gaussian being the "normal" case or the "one that usually matters". Finite variance is a big assumption and one that's routinely violated in practice.

For those that are interested, Levy-stable distributions are the general class of convergent sums of random variables [0], synonymously called "fat-tailed" or "heavy-tailed" distributions and include Pareto [1] and the Cauchy distributions [2].

Is there an intuitive explanation for why the Wigner semicircular law is basically the "logarithm" the Gaussian in some respect?

[0] https://en.wikipedia.org/wiki/L%C3%A9vy_distribution

[1] https://en.wikipedia.org/wiki/Pareto_distribution

[2] https://en.wikipedia.org/wiki/Cauchy_distribution