[Math] Are any linear combination of normal random variables, normally distributed

Question

It is easy to show that if we have n independent normally distributed random variables, then a linear combination fo them ar normally distributed.

It is also said that if (x1,x2,..,xn) is multivariate normally distributed, but not nececarrily independent, then any linear combination is also normally distributed. This is stated here: http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Definition

But does this mean that any linear combination of normally distributed random variables are normally distributed, even if they are not independent? This will follow from the definition if the joint distribution of set of normally distributed random variables(not nececarrily independent) are jointly multivarite distributed?

Answer 1

This will follow from the definition if the joint distribution of set of normally distributed random variables(not nececarrily independent) are jointly multivarite distributed?

Yes.

But does this mean that any linear combination of normally distributed random variables are normally distributed, even if they are not independent?

No. A frequently mentioned counterexample is based on $X$ standard normal and $Y=SX$ with $S=\pm1$ symmetric and independent of $X$. Then $X$ and $Y$ are normal but $X+Y$ is not since $P(X+Y=0)=\frac12$, a property that no normal random variable satisfies.