I have a question regarding the sum/linear combination of two multivariate Gaussian distributions, which are (not necessarily) jointly Gaussian. I mean, let's suppose that we have:
$$X\sim \mathcal{N}(\mu_1,K_{11})\\Y\sim \mathcal{N}(\mu_2,K_{22})$$
and $X,Y\in \mathbb{R}^{N}$. I am intereseted in the following linear combination of both random variables:
$$Z = a\odot X+b\odot Y$$
where $a,b\in\mathbb{R}$. If we cannot ensure that $X$ and $Y$ are jointly Gaussians, i.e $p(X,Y)\neq\mathcal{N}()$, can we know how to compute the covariance of $Z$? How would be the extension of a linear combination of $N$ random variables?
Just for completeness, can someone also provide the covariance when the joint distribution is jointly Gaussian, i.e $p(X,Y)=\mathcal{N}()$?
Thank you.