$Z=X+Y$,when $X$ and $Y$ are gaussian distribution ,must $Z$ be the gaussian distribution,too?

Question

Let $X$ and $Y$ be gaussian distribution,and $Z=X+Y$,must $Z$ be the gaussian distribution,too? my classmate said only when the covariance or correlation,I forgot,of $X$ and $Y$ be the same ,will the $Z$ be the gaussian too.

Can anyone correct me or give me some proof about it?

Answer 1

No, $Z$ need not be Gaussian. For example, let $X$ be Gaussian with mean zero and define $Y=-X$ if $|X|>1$ and $Y=X$ otherwise. Then $Y$ is also Gaussian (check its cdf), but $X+Y$ is clearly not Gaussian since it has a point mass at $0$ without being identically equal to $0$, and it is supported on $[-2,2]$ (whereas if it were Gaussian its range would be the whole real line).