TITLE: cumulant problem
QUESTION [6 upvotes]: 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.

REPLY [5 votes]: This quote might explain why your question on MSE went unanswered:

There is one obstacle that stands in the way of dealing with
  cumulants. We do not know simple necessary and sufficient inequalities
  on cumulants that ensure that a given sequence of real numbers is the
  sequence of cumulants of a random variable. For characteristic
  functions, such a necessary and sufficient condition is Bochner's
  theorem. To be sure, such inequalities exist in abundance if one
  expresses moments in terms of cumulants. The problem is finding the
  simplest such inequalities. The explicit determination of fundamental
  inequalities satisfied by the cumulants of a random variable is surely
  the most important open problem in probability theory.

Gian-Carlo Rota, 1998: Twelve problems in probability no one likes to bring up - Problem eleven: cumulants.
A status update has been given by Di Nardo and Senato, 2009, unfortunately behind a pay wall.