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Suppose that $X,Y$ are scalar random variables supported on some standard Lebesgue probability space $(\Omega, \mathrm{P})$, such that $X \overset{\mathrm{d}}{=} Y$ in the sense that their pushforward measures are equal, $X_*(\mathrm{P}) = Y_*(\mathrm{P})$. Does there exist a nondegenerate random variable $Z$ on $(\Omega, \mathrm{P})$ satisfying $X + Z \overset{\mathrm{d}}{=} Y + Z$?

In the case that $Y = X\circ T$ for some measure preserving automorphism $T: \Omega \rightarrow \Omega$ (this appears to be often the case by: Random variables with same distribution), and we have in addition the factorization $T = S^2$ for some automorphism $S: \Omega \rightarrow \Omega$, then clearly we can set $Z = X \circ S$, whence

$X + Z = X + X\circ S \overset{\mathrm{d}}{=} X \circ S + X \circ S^2 = Z + Y$.

Are there more general conditions than this, or perhaps conditions under which the answer to the question is negative?

Edit: it also turns out that if $Y = X \circ T$ where $T$ is measure preserving and non-ergodic then $Z$ exists; just take $Z$ such that $Z \circ T = Z$ almost surely and $Z$ non-constant. Then a computation like the one above shows equality in distribution.

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