On the origin of moment-cumulant formulas

When I gave a class on free probability theory a few years ago, I thought it would be a good idea to localize evidence for my usual statement that in the classical context the idea of viewing moment-cumulant formulas in terms of (multiplicative functions on) set partitions, as well as the vanishing of mixed cumulants in independent random variables, goes back to Rota; the main reference on this seemed to be the Foundations of Combinatorial Theory papers, part I and part VI. This is at least what I said in my old papers, like here or here, and what Jonathan Novak, for example, also iterates in his nice Three Lectures on Free Probability. But when I tried to find any mentioning of cumulants in those papers of Rota I could not localize anything. Also in the paper of Rota with Shen, On the Combinatorics of Cumulants, there is no clear mentioning of vanishing of mixed cumulants. I am still quite sure that I learned a lot and was inspired by the papers of Rota, but maybe this was more about multiplicative functions, and cumulants did not show up there explicitly. At this point I decided to ask Jonathan whether he has some clearer idea about the origin of the classical moment-cumulant formulas. Here is what he said:

About your question, I remember also having a difficult time tracking down a proof of the equivalence of independence and mixed cumulants vanishing in the literature.  I actually think that the earliest paper where this statement is explicitly made, with a complete proof given, is “Cumulants and partition lattices,” by T.P. Speed, Australian Journal of Statistics 25 (1983), 378-388. An annotated version of this paper appears in the collected works of Speed, edited by P. McCullagh (Chapter 6 of the volume). I hope this helps; I don’t know an earlier reference.

I was happy with this and more or less forgot about it. But a few days ago the same issue came up after a talk of Philippe Biane in the online seminar on Algebraic and Combinatoiral Perspectives in the Mathematical Sciences. It seems a couple of people are interested in this and could also provide a bit more information on aspects of the origin of moment-cumulant formulas, and maybe cumulants in general. So I thought it might be a good idea to collect here this information and invite others to add possibly some more remarks on this history.

Franz Lehner offered the following insightful remarks:

Here are some considerations concerning “Rota’s approach to cumulants”.

Both in his posthumous paper Rota/Shen: On the combinatorics of cumulants, J. Combin. Theory Ser. A 91 (2000), and in his Fubini lectures Twelve problems in probability no one likes to bring up he talks about the “Rota approach” but always with reference to Speed. So apparently he never published it himself explicitly, although he certainly knew it for a long time. Speed did not prove any new results, but gave elegant lattice theoretic proofs of known results (his notation is a bit messy though).

On the other hand it must be said that Rota did not invent the Möbius function either as he repeatedly mentions in his 1964 paper, but he clearly saw its fundamental importance (and proved some important results). Rota was a bird in the sense of Dyson and without his efforts to systematize and popularize it, the Möbius function would have remained in its oblivious state for yet another generation.

According to Rota, the Möbius function was invented by Weisner in the thirties. Multiplicative functions and the reduced incidence algebra were introduced in Doubilet/Rota/Stanley Foundations of Combinatorial Theory VI: the idea of generating function, 6th Berkeley Symposium on Probability, 1970/71. Cumulants are not mentioned there, but maybe not without reason this paper appeared in a Symposium on Probability. Similarly his 1964 paper on Möbius functions was not probabilistic, yet it appeared in Probability Theory and Related Fields. Again without mentioning cumulants explicitly, probably because “to prevent the length of this paper from growing beyond bounds, we have omitted applications of the theory”.

He just mentions in passing on p.359 that Schützenberger computed the Möbius function of the partition lattice (independently of Frucht and himself). Indeed in the cited paper Contribution aux applications statistiques de la théorie de l’information (Publ. Inst. Statist. Univ. Paris, 3(1-2):3–117, 1954, Thèse d’État) Schützenberger states as a remark on p.24 the Möbius formula for cumulants. To my knowledge this is the earliest occurrence; Leonov and Shiryaev also use the partition formalism in their 1959 paper, but apparently don’t know the concept of Möbius inversion.

In the statistics literature these developments went largely unnoticed for a long time and the graph theoretic calculation rules of Fisher, Kendall, James etc, remained the tool of choice, see the foreword by McCullagh to Speed’s collected works.

Joachim Kock added the following:

In Kendall’s ‘Advanced Theory of Statistics’ from 1945, there is already an explicit formula for cumulants in terms of moments, and the Möbius function (-1)^{n-1} (n-1)! appears explicitly in the formula! But of course, he doesn’t know that this combinatorial factor is the Möbius function.

In the notes he attributes various moment-cumulant relations to Frisch’s PhD thesis (Oslo 1926), but I don’t know if this particular formula is in there.

Regarding the Möbius function for posets, Weisner’s paper is from 1935 but it only deals with complete lattices, whereas Hall (independently) has the Möbius function for finite posets in his 1936 paper. In both cases, their proofs actually work the same for locally finite posets, which is Rota’s level of generality. (Stretching it a little bit, it is actually the same arguments that work for Möbius categories, and for certain abstract coalgebras.)

In case you are in a historical mood, allow me to advertise my paper From Möbius inversion to renormalisation. (It has no cumulants, though.)

Referring to the combinatorial factor, Franz could add some more insights:

Yes, this expression is already in Thiele’s 1899 paper (reprinted in Anders Hald The Early History of the Cumulants and the Gram-Charlier Series, International Statistical Review 68 (2000) 137-153, in English), but of course not realized as Möbius function, because that one was not known before Schützenberger.

It remains to clarify who was the first to explicitly write moments as a sum over set partitions. Leonov & Shiryaev just infer it from the factorial formula without comment and Schützenberger simply says “nous supposerons connu le fait”.

Thanks to Franz and Joachim for their remarks. Any more comments are welcome …

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