Comment by niemandhier
7 hours ago
Highly entertaining, here a little fun fact: there exist a generalisation of the central limit theorem for distributions without find out variance.
For some reasons this is much less known, also the implications are vast. Via the detour of stable distributions and limiting distributions, this generalised central limit theorem plays an important role in the rise of power laws in physics.
3blue1brown has a great series of videos on the central limit theorem, and it makes me wish there were something similar covering the generalised form in a similar format. I have a textbook on my reading list that covers it, unfortunately I'm I can't seem to find it or the title right now. (edit: it's "The Fundamentals of Heavy Tails" by Nair, Wierman, and Zwart from 2022)
Do you have any good sources for the physics angle?
https://en.wikipedia.org/wiki/Central_limit_theorem#The_gene...
Yes, came here to say the same thing. Telling people that the CLT makes strong assumptions is important.
Otherwise, they might end up underestimating rare events, with potentially catastrophic consequences. There are also CLTs for product and max operators, aside from the sum.
The Fundamentals of Heavy Tails: Properties, Emergence, and Estimation discusses these topics in a rigorous way, but without excessive mathematics. See: https://adamwierman.com/book
I thought the rise of power laws in physics is predominantly attributed to Kesten's law concerning multiplicative processes, e.g. https://arxiv.org/pdf/cond-mat/9708231
> find out
Finite?