Let $X$ and $Y$ be independent random variables and $f$ a smooth function. Is then f(X+Y) and X independent? Or under which conditions? Do you know where I can find something about this?
I know the proposition that $\phi_1(X), \phi_2(Y)$ are then independent for measurable $\phi$ but this doesn't seem to help. In the case of general $h(X,Y)$, the choice $h(X,Y)=X$ would proof it wrong, but this shouldn't be possible for h(X,Y) of the form f(X+Y)