Let $X$ and $Y$ be two real-valued random variables. Let $Z_{a,b}=aX+bY$ for $a,b\in\mathbb{R}$.
Suppose that for all $a,b\in\mathbb{R}$, the distribution of $Z_{a,b}$ is known. Then the joint distribution of $X$ and $Y$ is uniquely determined. This is because from the distributions of $Z_{a,b}$, we get the characteristic function of $(X,Y)$, which delivers the unique joint distribution.
My question is that, if we only know the distributions of $Z_{a,b}$ for some pairs of $(a,b)$, when can we recover a unique joint distribution of $(X,Y)$? Any reference for this type of question?
In general, we can't. For example if we only know the distributions of $Z_{1,0}=X$ and $Z_{0,1}=Y$, we can construct many different joint distributions.
But what would happen if we know the distributions of many $Z_{a,b}$? in particular, infinitely many? For example, suppose we know the distributions of $Z_{a,b}$ for all $a\geq 0$ and $b\geq 0$, can we recover the joint distribution? If not, what is the counter example?