# cumulant problem

A couple of days after I posted this to stackexchange, no one's answered:

I take the problem of cumulants to be this: given a sequence $(\kappa_1,\kappa_2,\kappa_3,\ldots)$ of real numbers, is it the sequence of cumulants of some probability distribution? In one sense, this is trivially equivalent to the problem of moments: the $n$th moment is a polynomial in the first $n$ cumulants and vice-versa. But cumulant sequences have a nice property that moment sequences don't have: the set of all such sequences is closed under addition. So draw a ray out from the origin $(0,0,0,\ldots)$. If the ray bumps into a cumulant sequence $(\kappa_1,\kappa_2,\kappa_3,\ldots)$, then $2(\kappa_1,\kappa_2,\kappa_3,\ldots),3(\kappa_1,\kappa_2,\kappa_3,\ldots),\ldots$ are also cumulant sequences.

For infinitely divisible distributions, for every real $t\ge 0$, the sequence $t(\kappa_1,\kappa_2,\kappa_3,\ldots)$ is a cumulant sequence.

Besides the nonnegative integers and the nonnegative reals, there are other sets of nonnegative reals closed under addition.

For which sets $T$ of nonnegative reals that are closed under addition is it the case that for some cumulant sequence $(\kappa_1,\kappa_2,\kappa_3,\ldots)$, for every real $t\ge 0$, $$t\in T \quad\text{iff}\quad t(\kappa_1,\kappa_2,\kappa_3,\ldots)\text{ is a cumulant sequence ?}$$

I'm guessing only closed sets, but there should be more than that to say about it, I would think. Even describing or classifying the (topologically) closed subsets of $\mathbb{R}^+$ that are closed under addition might be something of a chore, and I suspect many of them are not of this form.

That moderately complicated things might happen is at least hinted at by the case of $p\times p$ Wishart matrices, for which the number of degrees of freedom can be anywhere in the set $\lbrace 0,\dots,p-1\rbrace\cup (p-1,\infty)$. But that's for matrix-valued, rather than real-valued, random variables, so it's at most a hint.

• Can you include a link to the stackexchange question? Have you left a link there to the MO version? Mar 8, 2012 at 22:10
• Done. ${{{{{}}}}}$ Mar 8, 2012 at 22:29
• One condition mentioned on Wikipedia (the source of all truth), is that the cumulant generating function can't be a polynomial of degree greater than 2. (The normal distribution manages degree 2.) So you can't have finite number of non-zero cumulants except in some special cases. Mar 9, 2012 at 0:46
• ....and those special cases are the ones where all cumulants of degree $3$ or higher are $0$. If any cumulant of degree $\ge3$ is non-zero, then infinitely many are non-zero. However, this question is about the set of values of $t\ge 0$ for which $t\kappa$ is a cumulant sequence. Mar 9, 2012 at 18:39
• The moment problem is solved and understood to a certain extent. See arxiv.org/pdf/2008.12698.pdf. I'm also looking for if there's a nice translation in terms of cumulants though.. Nov 16, 2023 at 16:09

• QUOTE FROM THE FIRST PARAGRAPH: One of many mistakes of my youth was writing a textbook in ordinary differential equations. It set me back several years in my career in mathematics. However, it had a redeeming feature: it led me to realize that I had no idea what a differential equation is. The more I teach differential equations, the less I understand the mystery of differential equations. END QUOTE $\qquad$ Apr 20, 2017 at 15:01
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