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The question goes, if $X, Y, Z$ are independent random variables, where $Z$ is normal, and $X+Z, Y+Z$ have the same distribution, show $X, Y$ have the same distribution. I don't know whether and how generating functions can help, because $X+Z$ and $Z$ are not independent, and I cannot argue by subtracting $Z$ from $X+Z$. Could anyone please give any hint?

More generally, is it true when we don't assume $Z$ is normal?

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    $\begingroup$ This is not true. in the general case. There is a counter-example in Feller's book. $\endgroup$
    – geetha290krm
    Nov 2, 2022 at 5:06

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An easy proof when $Z$ is normal is using characteristic functions. Let $f,g,h$ be the characteristic functions of $X,Y$ and $Z$ respectively. Then $f(t)h(t)=h(t)g(t)$ and $h(t)\neq 0$ for all $t$ so $f=g$ which implies that $X$ and $Y$ have the same characteristic function.

This is not true. in the general case. See Curiosities ii) in under "Special densities: mixtures"in Vol II (Ch. on characteristic functions) in Feller's book.

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  • $\begingroup$ So normality of $Z$ can be replaced with "has a nonzero characteristic function"? $\endgroup$
    – angryavian
    Nov 2, 2022 at 15:22
  • $\begingroup$ @angryavian Yes, that is correct. $\endgroup$
    – geetha290krm
    Nov 2, 2022 at 23:15
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If $X$ and $Y$ have the same distribution, can you show that $X + Z$ and $Y + Z$ have the same distribution for $Z$ normal with $X,Y$ and $Z$ independent? Call this Claim 1.

If so, since $X + Z$ and $Y + Z$ have the same distribution, add the normal distribution $-Z$ to $X + Z$ and $Y + Z$, by Claim 1 the resulting distributions are the same (namely, $X$ and $Y$ are the same).

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