How can I prove $Y=X+Z$ is a Gaussian Random Variable iff $Z$ is a Random Variable?

Question

Let $X$ and $Y$ be Gaussian Variables. We know $Y=X+Z$.

Let $X$ and $Z$ be independent. How can I prove $Y$ is a Gaussian Random Variable iff $Z$ is a Random Variable?

Can I use $X$, $Z$ Orthogonal and Normal thus create a Gaussian Vector hence any Linear Combination is a Gaussian Variable?

Answer 1

If $X$ and $Z$ are independent random variables and $X+Z$ is a Gaussian random variable, then $X$ and $Z$ are Gaussian random variables too.

This is a result in Feller's book where it is attributed to a conjecture by P. Lévy that was proved by H. Cramér. See this answer for more details. The proof of the general statement above is not easy. But, if you know that $X$ is Gaussian (note that this is not part of the assumptions in the statement above), then, given that $Y=X+Z$ is Gaussian, there is an easy proof that $Z$ is also Gaussian via characteristic functions as suggested in ChargeShiver's comment.