[fgimenez]

Thanks for this cool writeup. My undergrad probability professor had us prove the CLT using cumulants, but I never kept the notes and had trouble finding the same proof later in life. Do you know of any reference that goes in depth about cumulants/FT/CLT?

[drostie]

Actually probably one of the best simple ones is Scholarpedia's article: in particular it covers one of the things that are kind of at the periphery of a non-applied statistician's mind when using cumulants: how the log-normal distribution serves as a handy counterexample to just about everything you'd like to prove about them (e.g. with some tweaking you can prove that the log-normal distribution's cumulant expansion is not unique, therefore you have an injection from PDFs to cumulants, not a surjection).

Something I've never seen rigorously explored: Cumulants appeared in my Master's work in condensed matter, where they at times seemed to have a mysterious tie to Feynman diagrams; that is, there seems to be some sort of analogy between how the Nth cumulant "subtracts out" all of the components of the Nth moment which are just products of the previous N-1 moments to obtain something "additive", and a Feynman diagram containing a vertex joining N particles. In particular Feynman diagrams for pairwise interactions seemed to be eerily similar to cumulant expansions which did not have any 3rd or higher moments. I confess I totally forgot this until I Googled today and saw a question on Math Overflow about it, but it was too sketchy to show me a real satisfying resolution of my mental curiosity. Similarly I remember seeing Young tableaux in some lecture notes on cumulants and then the day after I was working with Marcin Dukalski on some crazy approach he was taking in his thesis, and he spontaneously started talking about "hm, what do the Young tableaux look like for that?" and I was only able to help at all because of some limited understanding from the previous day's reading! So... there's definitely some mathematical resonances when you get to quantum theory.