After attending a recent talk at TTI about dimension reduction by Moses Charikar in which he mentioned the special role stable distributions play, I made a note to freshen up my own scattershot knowledge of facts about stable distributions. Of course, things got too busy and the the note ended up on my sub-list of to-do items that get infinitely postponed. However, I’ve been saved by a recent post to ArXiV by Svante Janson, who does all sorts of interesting work on these cool objects called graphons (the limits of infinite graph processes) :
Stable Distributions
Svante JansonWe give some explicit calculations for stable distributions and convergence to them, mainly based on less explicit results in Feller (1971). The main purpose is to provide ourselves with easy reference to explicit formulas. (There are no new results.)
All (or at least most) of the facts I wanted in one place! Hooray!
He starts with infinitely divisible distributions (e.g. Gaussian, Poisson, Gamma) and then talks about 
![\alpha \in (0,2]](https://s0.wp.com/latex.php?latex=%5Calpha+%5Cin+%280%2C2%5D&bg=ffffff&fg=333333&s=0&c=20201002)
An Ergodic Walk 
I have been using the document above recently. In one of my projects, I found that a stable distribution was a good functional fit to a signal. Just wanted to learn more about the origin. Very nice ! The surprising thing to me, is that there exists stable distributions for very small alphas… Although the numerical solvers to obtain their actual shapes become super slow.