Sum of two (real or not?) gaussian random variables gives a gaussian random variable

I'd like to know if, given two normal random variables $X$ and $Y$, the new random variable $Z=X+Y$ is normal as well. Is it necessary that $X$ and $Y$ are real normal random variables or is it sufficient that they are normal random variables?

EDIT: $X$ and $Y$ are dependent and $(X, Y)$ is a gaussian random vector.

Solution 1:

In general, the sum of normal random variables is not normal (unless they are independent). See here for an example. However, if they are jointly (multivariate) normal, then any linear combination of them will be normal. In fact, when any linear combination of the components of a random vector is normal, this is one definition of being multivariate normal.

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