Independence of the sum of random variables

Question

Let $X,Y,Z$ be random variables. Is it true that X is independent of Z if and only if X+Y is independent of Z+Y? I suspect not, because if we have Y = -X, X+Y is no longer a random variable. Does the statement become true if X+Y and Z+Y are both nondeterministic random variables?

Answer 1

Let $X=Z=0$ and $Y$ be a non-degenerate random variable (e.g. $\text{Bernoulli}(1/2)$). Then $X$ and $Z$ are independent but $X+Y=Y$ and $Z+Y=Y$ are dependent.

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If you insist on an example with all non-degenerate random variables:

Let $X$ and $Z$ be independent $\text{Bernoulli}(1/2)$. Let $Y$ take values $100$ and $-100$ each with probability $1/2$, also independent of $X$ and $Y$. Then $X$ and $Z$ are independent by construction. But $$P(X+Y=100) = P(X=0, Y=100) = 1/2^2$$ $$P(Z+Y=100) = P(Z=0, Y=100) = 1/2^2$$ but $$P(X+Y=100, Z+Y=100) = P(X=0, Z=0, Y=100) = 1/2^3 \ne (1/2^2)^2.$$