Suppose I have $N$ random variables $X_1,\dots,X_N$. I do not know the joint distribution.
However, I do know the marginal distribution $f_i$ of each $X_i$, and I also know the marginal distribution of all possible sums, i.e., I know the marginal distribution $g_S$ of $\sum_{i \in S} X_i$ for any subset of $S\subseteq\{1,\dots,N\}$.
What do I learn about the joint distribution? (I know this is a vague question as I don't specify what exactly I mean). Can I fully back out the joint distribution (counterexample or proof)? Do I learn anything about the correlation of the random variables in some $S$?