Consider four random variables $X_1, X_2, X_3, X_4$. Let $X_1$ and $X_2$ be independent as well as $X_3$ and $X_4$. We also have that $X_1+X_2$ and $X_3 + X_4$ are independent. Does this imply that all four random variables are independent? If not, is there a simple counter-example?
I suspect that they are not independent but I have failed to come up with a counter-example.
If we only have that $X_1+X_2$ and $X_3 + X_4$ are independent then $X_1 = -X_2 = X_3$ is an example where $X_1$ and $X_3$ are not independent but $X_ 1+X_2$ and $X_3 + X_4$ are independent.