It is well known that if $X$ and $Y$ are two independent gaussian random variables, or if $(X,Y)$ is jointly bivariate gaussian, then the sum $X+Y$ is a gaussian random variable. It is also known that without any of such assumptions, the conclusion may fail.
It is also known that if $X$ and $Y$ are independent random variables such that the sum $X+Y$ is gaussian, then $X$ and $Y$ are gaussian variables.
The question is: is it possible to find an example of two random variables correlated $X$ and $Y$ (not constant) whose sum $X+Y$ is Gaussian (not degenerate), but such that $X$ or $Y$ is not gaussian?
The question is: is it possible to find an example of two random variables correlated and (not constant) whose sum + is Gaussian (not degenerate), but such that or is not gaussian?
Consider standard normal $Z$
$X :=\max(0,Z)$
$Y:=\min(0,Z)$
$X$ is bounded below by a constant (0) and $Y$ is bounded above by a constant (0) so neither is Gaussian.
then $X+Y = Z$, and $E\big[X\big] = -E\big[Y\big] \neq 0$, and
$Cov(X,Y) = E\big[XY\big] -E\big[X\big]E\big[Y\big] = 0 -\Big(-E\big[Y\big]\Big)E\big[Y\big] = E\big[Y\big]^2 \gt 0$