Q:

Sum of dependent normal random variables

Let ${\bf X} =(X_1,\ldots,X_n)'$ be a vector of random variables that may be dependent and let ${\bf a}=(a_1,\ldots,a_n)'$ and ${\bf b}=(b_1,\ldots,b_n)'$ be nonrandom vectors with $a_i \neq 0$ and $b_i \neq 0$ for $i=1,\ldots,n$. 
Assume $ {\bf a' \bf X} =\sum_{i=1}^n a_i X_i \sim N(0,\sigma_a^2)$ and $ {\bf b' \bf X} =\sum_{i=1}^n b_i X_i  \sim N(0,\sigma_b^2)$ be normal random variables. Is $ {(\bf a' + \bf b') \bf X}$ also a normal random variable?

A:

Not always--otherwise every sum of normal random variables would be normal, and this ain't so.
Canonical (counter)example: Assume that  $\xi$ is standard normal and that $\eta=\sigma\xi$, where $\sigma=\pm1$ is symmetric Bernoulli and independent of $\xi$. Then $\eta$ is standard normal but $\xi+\eta$ is not normal since $P(\xi+\eta=0)=P(\sigma=-1)=\frac12$ while $P(\zeta=0)$ is $0$ or $1$ for every normal random variable $\zeta$. (This argument proves that the vector $(\xi,\eta)$ is not normal.)
Variant of the same: $X=(\xi,\xi,\sigma\xi,\sigma\xi)$, $a=(1,1,1,-1)$, $b=(1,-1,1,1)$.

