$X,Y,Z$ are iid random variables $\implies$ $X-Y$ are $Y-Z$ are independent

Question

Let $X,Y,Z$ be independently and identically distributed random variables. Is it true in general that $X-Y$ and $Y-Z$ are independent$?$ Is there any counter example$?$

Suppose that $X, Y, Z$ are independent and that $X-Y$ and $Y-Z$ are independent. These two conditions necessarily imply that $Y$ is a constant random variable. This can be easily read out from Robert Israel's answer. — Sangchul Lee, Dec 25 '17 at 07:21

score 1 · Answer 1 · answered Dec 25 '17 at 07:16

1

No, it's not true. Any nontrivial example you try will be a counterexample, because $\text{Cov}(X-Y, Y-Z) = - \text{Var}(Y)$.

answered Dec 25 '17 at 07:16

Robert Israel

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$X,Y,Z$ are iid random variables $\implies$ $X-Y$ are $Y-Z$ are independent

1 Answers1