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Near the end of "Finite Operator Calculus" (1976), G.C. Rota writes:

Note that one can define cumulants relative to any sequence of binomial type, e.g. the factorial cumulants (Kendall and Stuart).

The book he makes reference to is listed in the bibliography as "M. G. Kendall and A. Stuart, “The Advanced Theory of Statistics”, Vol. I, Griffin, London, 1963". I was only able to find the 1945 version of the book in abysmal quality, and I wasn't able to find any reference to "factorial cumulants" there (for instance, there is no reference to "factorial cumulants" or "cumulants, factorial" in the index). Perhaps, they were added in the 1963 edition.

I am in a great need to understand this phrase by Rota, since it may have a direct connection to my current research topic, so any references or explanations would be greatly appreciated, especially if there is an established direct connection with free probability.

My own guess is the following (based on the definition of factorial cumulants found in this article: Kambly, Flindt, and Büttiker - Factorial cumulants reveal interactions in counting statistics. Let $b_{k}(z)$ be a polynomial sequence of binomial type (for instance, lower (falling) factorial sequence, as in the article). Define as $$M_{b}(z)[X]:=\sum_{k=0}^{\infty}\frac{\mathbb{E}[b_{k}(X)]}{k!}z^k$$ the moment generating function of $X$. Then define $$S_b(z)[X]=\ln M_{b}(z)[X]$$ as the new cumulant generating function, such that $$\frac{d^{k}}{dz^{k}}\Biggr|_{z=0}S_{b}(z)[X]=\kappa_{b}(X)$$ and name these new objects cumulants associated to a given binomial type sequence.

I suppose, the reasoning behind this is the following: let $M_{b}(z)[X], M_{b}(z)[Y]$ be generating functions for the same binomial type sequence. Then their product is \begin{gather*} M_{b}(z)[X]M_{b}(z)[Y]=\sum_{n=0}^{\infty}z^{n}\sum_{k+l=n}\frac{\E[b_k(X)]\E[b_{l}(Y)]}{k! l!}= \\ =\sum_{n=0}^{\infty}\frac{z^n}{n!}\sum_{k=0}^{n}{n \choose k}\E[b_k(X)]\E[b_{n-k}(Y)]. \end{gather*} If we denote $m_{j}(X)=\E[X^j]$, then $\E[b_k(X)]=\sum_{l}{b_{kl}m_{l}(X)}$ where $b_k(x)=\sum_{l}b_{kl}x^l$, and $m_{k}(X)$ is itself a binomial type sequence of sort, in the sense that for independent $X$, $Y$ $$m_{n}(X+Y)=\sum_{k=0}^{n}{n \choose k}m_{k}(X)m_{n-k}(Y)$$ so $\E[b_{k}(X)]$ is an "umbral composition" $b_{k}(\vec{m}(X))$. Then \begin{gather*} M_{b}(z)[X]M_{b}(z)[Y]=\sum_{n=0}^{\infty}\frac{z^n}{n!}\sum_{k=0}^{n}{n \choose k}b_{k}(\vec{m}(X))b_{n-k}(\vec{m}(Y))\cong \\ \cong \sum_{n=0}^{\infty}\frac{z^n}{n!}b_{n}(\vec{m}(X+Y))=M_{b}(z)[X+Y]. \end{gather*}

(I write $\cong$ in the last transition because I am not entirely sure in the whole calculation and reasoning, and not yet comfortable with using umbral calculus language.) If this reasoning is correct, then the virtue of generalized binomial type moments is the following: just like for regular moments, sum of independent random variables corresponds to multiplication of their moment generating functions (and, I guess, binomial type cumulants linearize their addition).

However, these are my speculations, and I have not found any literature on the topic. Could anyone help with references?

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    $\begingroup$ MathJax note: commands like $\newcommand\E{\mathbb E}$ are eaten up with other white space by TeX, but not by MathJax. To avoid creating spurious extra lines in the post, the closing or ending $ must be on the same line as surrounding text (although white space inside $ $ is eaten as one would expect). I edited accordingly. \\ Your source refers to $M_b(z)$ as the <binomial sequence> moment generating function of $X$. Here, I think that <binomial sequence> is taken as malformed HTML, and discarded. Was that intentional? $\endgroup$
    – LSpice
    Commented Jan 18 at 0:52
  • $\begingroup$ Thank you for the advices! I do not understand your question though. Don't remember putting anything in < > brackets. $\endgroup$ Commented Jan 18 at 14:01
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    $\begingroup$ Re, it may have been a typo. But, if you look at the original source, then you will find the phrase <binomial sequence> moment generating function of $X$ that I have indicated, which is still in the latest revision. If the rendered result is as you expect, then all is well! But, if the word "binomial sequence" is missing where you expect it, then that is why. $\endgroup$
    – LSpice
    Commented Jan 18 at 15:15
  • $\begingroup$ Another MathJax note: It ought to say \newcommand\E{\operatorname{\mathbb E}}, in order to get context-dependent spacing that \operatorname causes. $\endgroup$ Commented Jan 24 at 21:37

3 Answers 3

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Suppose a sequence $\big( p_n(a) \big)_{n=0}^\infty$ of polynomials with real coefficients satisfies

  • $\deg p_n(a) = n,$
  • $\displaystyle p_n(a+b) = \sum_{k=0}^n \binom n k p_k(a) p_{n-k}(b)$ for every $n.$

That is what it means to say that this sequence is of binomial type. (One can greatly weaken the assumption that the coefficients are real and this definition remains intact, but here we will be concerned with the case of real coefficients.)

In every such sequence one has $p_0(a)=1$ and $p_n(0)=0$ for $n>0.$

Lemma: Let $c_1,c_2,c_3,\ldots$ be any sequence of scalars. Then there is exactly one polynomial sequence $\big( p_n(a) \big)_{n=0}^\infty $ of binomial type for which $p_n'(0)=c_n$ for every $n\ge 1.$

This can be proved by induction on $n.$

The $n\text{th}$ moment of the probability distribution of a random variable $X$ is $\operatorname E\left( X^n\right).$

The $n$th cumulant $\kappa_n(X)$ of the distribution is a polynomial function of the first $n$ moments, with the properties that

  • one term of that polynomial is just the $n$th moment, with coefficient $1$, and
  • $\kappa_n(aX) = a^n \kappa_n(X)$ for constant (i.e. non-random) $a$ ($n$th-degree homogeneity), and
  • if $X_1,\ldots, X_m$ are independent, then $\kappa_n(X_1+\cdots+X_m) = \kappa_n(X_1) + \cdots + \kappa_n(X_m).$

(E.g., the second and third cumulants are just the central moments, and the fourth cumulant is the fourth central moment minus three times the square of the variance.)

Let $c_1,c_2,c_3,\ldots$ be the cumulants of some probability distribution (for which all moments are finite). Let $\big( p_n(a) \big)_{n=0}^\infty$ be the sequence of polynomials in the lemma above. Then $p_n(a)$ is the $n$th moment of the distribution of the sum of $a$ i.i.d. copies of a random variable with that probability distribution. Or if you like, it is the $n$th moment of the $a$th convolution-power of that distribution. And in case the distribution is infinitely divisible, the analogous statement is true of non-integer positive values of $a.$

Thus \begin{align} \operatorname E\big( (X_1+\cdots + X_a)^n \big) & = p_n(a), \tag 1\label{462420_1}\\[8pt] \kappa_n(X_1+\cdots + X_a) & = ap_n'(0). \tag 2\label{462420_2} \end{align}

So what happens if instead of the expected value of $X^n$, which is the “$n\text{th}$ moment of $X$ relative to the polynomial sequence $\bigl( a^n \bigr)_{n=0}^\infty$of binomial type,” we put some other sequence of binomial type here? That gives us moments relative that other sequence of binomial type, and the cumulants would be related to the moments just as in lines \eqref{462420_1} and \eqref{462420_2} above. I would guess that that's what Rota had in mind.

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I am not quite sure which generalization Rota had in mind, but this is a generalization that has found applications in the literature on stochastic processes:

Given a probability distribution function $P(n)$ for a discrete variable $n\in\mathbb{N}$, construct the generating function $M(z)=\sum_{n=0}^\infty z^nP(n)$ and obtain the generalized cumulants $$\kappa_p(s)=\left.\frac{d^k}{dz^k}\log M(z)\right|_{z=s}.$$ Counting factors enter the generating function as powers of $z$ and not $e^z$ as in the case of ordinary cumulants. The factorial cumulants correspond to $s=1$.
The construction is particularly useful if a certain distribution has only a few nonzero generalized cumulants (generalizing the Gaussian distribution for ordinary cumulants and the Poisson distribution for factorial cumulants).

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  • $\begingroup$ I remember vaguely that in umbral calculus, all sequences of binomial type are parametrized by a choice of substitution parameter, looks interesting, thanks $\endgroup$ Commented Jan 17 at 17:53
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Rota and Shen published "On the Combinatorics of Cumulants" in 2000, but no mention of factorial cumulants is made. (Beware: my notation is different from their's. I use the subscript, or index, '.' simply to flag an umbral character. They use it as defined at the bottom of p. 287.)

I was able to view Kendall and Stuart's The Advanced Theory of Statistics Vols. 1 & 2, but found only discussions of product-cumulants and product-moments with which I'm not familiar. Searching further on 'product-cumulants and Kendall', I found the paper "The derivation of multivariate sampling formulae from univariate formulae by symbolic operation" by Kendall, in which he shows the standard relationship--the exp-log relationship--between the classical formal cumulants and formal moments, resulting in the inverse pair of partition polynomials the Faa di Bruno / Bell / exponential composition polynomials of OEIS A036040 and the cumulant expansion polynomials of A127671 / A263634 (see pages 394-395 in Kendall's paper for a partial listing of these two sets or the Lang links in the OEIS entries). So, I thought I would review and provide links for the classical and free cumulants and some of their relations with the hybrid umbral-finite operator calculus of Sheffer polynomial sequences.

The Bell polynomials provide the expansion

$$e^{xg(t)} = e^{\operatorname{Bell.}(a_1,a_2,\ldots;x)t}$$

where, with $(a.)^0 = a_0 = 0$,

$$g(t) = e^{a.t} = a_1t + a_2 \frac{t^2}{2!} + a_3 \frac{t^3}{3!} + \cdots,$$

a formal e.g.f..

A Taylor series expansion gives

$$e^{\operatorname{Bell.}(c_1,c_2,\ldots;x)t} = e^{x (c_1t + c_2t^2/2! + c_3t^3/3!+...)}$$

$$=1 + c_1 x t + x (c_1^2 x + c_2)t^2/2! + x (c_1^3 x^2 + 3 c_2 c_1 x + c_3) t^3/3! + x (c_1^4 x^3 + 6 c_2 c_1^2 x^2 + 4 c_3 c_1 x + 3 c_2^2 x + c_4)t^4/4! + O(t^5),$$

where, e.g., with $x=1$,

$$m_3 = \operatorname{Bell}_3(c_1,c_2,c_3;1)= c_1^3 + 3 c_2 c_1 + c_3$$

is the formal moment $m_3$ in terms of the formal cumulants $c_1,c_2,c_3$.

Conversely, formally

$$c_1t + c_2t^2/2! + c_3t^3/3!+ \ldots = \ln[e^{x (c_1t + c_2t^2/2! + c_3t^3/3!+...)}]|_{x=1}$$

$$= \ln[e^{\operatorname{Bell.}(c_1,c_2,\ldots;1)t}]=\ln(1 + m_1t+m_2t^2/2! +m_3t^3/3! + \cdots)$$

$$=m_1 t + (m_2 - m_1^2) t^2/2! + (2 m_1^3 - 3 m_2 m_1 + m_3) t^3/3! + (-6 m_1^4 + 12 m_2 m_1^2 - 4 m_3 m_1 - 3 m_2^2 + m_4) t^4/4! +O(t^5) ,$$

and, e.g.,

$$c_3 = 2 m_1^3 - 3 m_2 m_1 + m_3$$

is the third cumulant $c_3$ in terms of the moments $m_1,m_2,m_3$. These are known as the cumulant expansion polynomials or logarithmic polynomials.

The e.g.f. defining the general Bell polynomials is precisely of the form of the e.g.f. of a generic binomial Sheffer polynomial sequence in the variable $x$ with $a_1 \neq 0$. Rota said in the passage you mentioned "Note that one can define cumulants relative to any sequence of binomial type", implying that the existence of an underlying probability distribution is irrelevant / not required (this is repeated in Rota and Shen). The algebra is graded, so even convergence of the e.g.f.s is not required. The Bell polynomials with $x=1$ are also an Appell Sheffer sequence in the distinguished indeterminate $a_1$, as the derivative at the top of p. 394 of Kendall implies, and, therefore, subject to the umbral maneuvers and op calculus of Appell as well as binomial Sheffer sequences.

The cumulant expansion polynomials, as discussed in A263634, are intimately linked to the Appell Sheffer finite op calc / umbral calculus as well. As noted in A127671, they are also intimately related to the Faber polynomials of A263916 of symmetric function theory, which satisfy a slew of algebraic and differential identities.

Senato and Di Nardo discuss boolean, classical, and free probability moments and cumulants in several papers with a stronger flavor of probability theory from the perspective of the hybrid umbral-finite operator calculus, e.g., P1, P2. (They speak of factorial moments.)

The free cumulants and moments of free probability theory are related to the Kreweras-Voiculescu polynomials of A134264, modeled by a number of geometric combinatorial constructs, including noncrossing partitions of polygons, trees, Dyck paths, and parking functions. The inverse set of partition polynomials are those of A350499. Some relationships between these two sets and compositional inversion of Laurent series are given in my MO-Q "Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory". A great introduction to both classical and free cumulants is "Three lectures on free probability" by Novak and LaCroix. A relationship between umbral calculus and the formation of the K-V partition polynomials is sketched in A134264. The K-V polynomials are an Appell Sheffer sequence in a distinguished indeterminate, as I noted in my OEIS entry and my contribution to your MO-Q "Are umbral moonshine and umbral calculus connected?".

You might also like to scan my answer and comments to the MO-Q "Meaning of a quote of Doubilet, Rota and Stanley on harmonic analysis and combinatorics".

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