This question came to my mind when I was reading a theorem on Poisson Processes stating that the sum of two independent Poisson processes, $X_t$ and $Y_t$ is a Poisson process. In the proof, one has to show that $X_t + Y_t$ has independent increments, and in order to prove this one must show the question I posed. Is it true? Are there any counterexamples?
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Try $(X,Y,Z)$ uniform on ${(0,0,0),(1,1,0),(1,0,1)(0,1,1)}$. – Did Dec 09 '18 at 14:09
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I am not concerned to prove that they are not mutually independent. I just need in the theorem to deduce in some way that X is independent of Y+Z, or maybe of the vector (Y,Z). otherwise I cannot see how the proof of the theorem can be continued. – Petros Karajan Dec 09 '18 at 14:54