I want to show that the sum of two independent gaussian vectors is a gaussian vector. We had, that a gaussian vector can be written as $X=A*Z+b$ where $A$ is a real matrix, $b$ is a real vector and $Z$ is a vector with standardnormal distributed components. So let $X$ and $Y$ be two n-dimensional gaussian vectors. So we can write them in the following way:
$X=A*Z+b,~Y=B*Z+c,~ A,B \in \mathbb{R}^{nxn},~ c,b \in \mathbb{R}^n,~ Z \in \mathbb{R}^n,~Z\sim N(0,I)$
Can I use the same $Z$ here? Then the sum would be
$X+Y=(A+B)*Z+(b+c)$
and so it would be a gaussian vector. Is that correct or do I have to argue in a different way?
And later I have to find out the covariance matrix, can I argue that cov($x_i, y_j$)=0, because the vectors $X$ and $Y$ are independent?
Thank you! Susan