Hey, I'm new here and this is my first attempt at a question. My background is in econometrics, so I apologize in advance if I use unfamiliar notation or display ignorance of important work.

Suppose we observe a large amount of data $\{(Z_{i})\}_{i=1}^{n}$

generated i.i.d. by $Z_{i }= X_{i} + Y_{i}$ with $Y$ representing contaminating noise with known distribution. Interest is in working backwards (*deconvolution*) to learn (*identify*) as much as possible about the distribution of $X$. Let the characteristic functions of these variables be denoted as $\phi_{X}(t) \equiv E(e^{ixt})$ and similar notation for $Y$ and $Z$.

It is well known that the full distribution of $X$ can be learned under the restriction that $X \perp Y$. This follows because $\phi_{Z}(t)$ can be consistently estimated from the data and $\phi_{X}(t) = \frac{\phi_{Z}(t)}{\phi_{Y}(t)}$. However, I am interested in relaxing this restriction and replacing it with the weaker requirement that $\forall x\in \text{Supp}(X)$, $E(Y|X=x)=0$.

Under this restriction, I know that (for example), the variance of the unknown distribution $X$ can be learned by:

$Cov(X,Y)=0 \implies Var(X) = Var(Z) - Var(Y).$

However, I am fairly certain that I cannot learn the entire distribution of $X$ with this weaker assumption. If you can think of a good counter-example, that would be helpful. More importantly, I would like to make the strongest possible statements about the full distribution of $X$ based on knowledge of the distributions of $Y$ and $Z$.

Edited to add-----------

I have become aware that if I altered my assumption to require that $med(Y|X)=0$ I would pin down a horizontal section of the copula between Y and X and that the bounds in that case have been well studied (for example: tandfonline.com/doi/abs/10.1080/03610920701386976). However, this is a less useful result. Perhaps no aspect of a copula is pinned down by a conditional mean restriction? Would this limit my ability to provide meaningful bounds on quantiles of the $X$ distribution?