TITLE: Why a random variable is better described by its cumulants than by its characteristic function?
QUESTION [5 upvotes]: It is a classical and well known problem that a random variable $X$ is not uniquely determined by its moments $\mathbb{E}(X_n)$. The moment problem is the problem of determining the probability density of a random variable in terms of its moments, as well as the uniqueness of such a density given by the moments.
I have read this from the Algebraic Combinatorics and Computer Science of H. Crapo and D. Senato:

"In the first half of this century, the method of moments was replaced by Paul Levy by a more pliable method relying upon the characteristic function $\mathbb{E}(\mathrm{e}^{\mathrm{i}tX })$, which is used to this day to derive the limit theorems of probability in their sharpest form. There is however one drawback to the characteristic function: it has no obvious probabilistic significance. My teacher William Feller was aware that the religious invocation of characteristic functions is extraneous to probabilistic reasoning. He managed to avoid characteristic functions in his treatise on probability. To be sure, characteristic functions made an occasional appearance in the second volume, when he just could not do without them. However, it was his intention to write a third volume, dealing with Brownian motion and diffusion processes, in which characteristic functions would be relegated to the dustbin of history. Unfortunately, he died before he could accomplish this task."

Other important objects that can better represent many properties of random variables than  moments are cumulants.
For $n\geq 1$, we consider a vector of real-valued random variables $X_{[n]}= (X_1,\ldots, X_n)$ such that $\mathbb{E}(|X_j|^{n}) <\infty, \forall\  j = 1,\ldots,n$. For every subset $b = \{j_1,\ldots, j_k\} \subset [n]=\{1,\ldots,n\}$ , one writes
$X_b=(X_{j_1},\ldots, X_{j_k})$ and $X^{b}=X_{j_1}\times \ldots\times X_{j_k}$. From the $k$-dimensional vector $X_b$,
one cane define the multivariate characteristicfunction of this vector as
$$\phi_{X_b}(z_1,\ldots,z_k)=\mathbb{E}\Bigg[\exp\Big(\mathrm{i}\sum_{l=1}^{k}z_l X_{j_l}\Big)\Bigg].$$
The joint cumulant of the components of the vector $X_b$ is defined as
\begin{align*}
k(X_b)=(-\mathrm{i})^{k} \frac{\partial^{k} }{\partial z_1\ldots\partial z_k}\log\phi_{X_b}(z_1,\ldots,z_k)|_{z_1=\ldots=z_k=0}.
\end{align*}
I know that cumulants are polynomials in moments, invariants by the translation and additive if the the components of the vector $X_b$ are partly independents.
They have also common properties with characteristic and generating function in the sense they characterize uniquely the distribution of a random variable as also they can characterize the independence of random variables.
My question: why is a random variable is better described by its cumulants, which are combinatoric nature, than by its generating function or its characteristic function, which are analytical nature.

REPLY [3 votes]: I think it is a bit misleading to contrast cumulants as combinatorial quantities with the characteristic function as analytic object. The characteristic function is an analytical device containing information about the moments. In the same way there is an analytic object (namely the logarithm of the characteristic function) which contains information about the cumulants. So I would say that there are moments and there are cumulants, dealing with them has often a combinatorial flavor, and there are analytic reformulations of moments and of cumulants which allow the use of more analytic tools (and, in particular, allow to deal with situations where no moments/cumulants exist). In some cases moments are better suited for the problem at hand, in some cases cumulants are.
In the classical case, the closeness between the analytic avatars of moments and of cumulants (the first is the characteristic function, the second is the logarithm of the characteristic function) might
be a reason that one usually does not talk so much about the analytic version of cumulants. In free probability theory the difference between the Cauchy transform (the analytic function for moments) and the $R$-transform (the analytic function for free cumulants) is much bigger
and the parallelism between the combinatorial and the analytical side of moments/cumulants is more visible.