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."
Others 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 is to know why 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.
