Moment Generating Function with Jointly Gaussian Vector

Question

The goal is to prove that for $X$ a jointly Gaussian random vector and $A \in \mathbb{R}^{M \times N}$ a rank $M$ matrix that $Y=AX$ is also jointly Gaussian.

I've found that a common way to prove this is using the moment generating function. Here to save space, we'll let $t = (t_1, \ldots, t_n)$. So $M_Y(t) = E[e^{t'Y}] = E[e^{t'AX}]$. What I'm struggling with is what to do next.

Any hints or suggestions would be much appreciated.

Answer 1

A random variable is multivariate gaussian iff the moment generating function looks a certain way:

$M_X = \exp(\mu t + 0.5t'\Sigma t)$

(see also here https://en.wikipedia.org/wiki/Multivariate_normal_distribution )

So you know already how $M_X$ looks. Now try to prove that $M_Y = M_{AX}$ can be described in the same way with some $\tilde{\mu}, \tilde{\Sigma}$.

Answer 2

You could write $t'AX = (A't)'X$, so $M_Y(t) = M_X(A't)$