Are linear combinations (i.e. "sums") of gaussian distributions also gaussian? I have always assumed that this is fact : but are there any mathematical theorems that state: finite sums of gaussian distributions are gaussian themselves?

In the above picture, does p(x) have a gaussian distribution? Do the sums need to be linear? Can they be non-linear?
I saw the following question: Is the joint distribution of two linear combinations of Gaussians still a multivariate normal? - but no reference/proof was provided.
Thanks

Suppose that you have two random variables $x\sim F_x$ and $y\sim F_y$ where $F_x,F_y$ are probability distributions. Then generally $w_xx+w_yy$ is not from $w_xF_x+w_yF_y$, even if when $F_x,F_y$ are normal distributions.
The characteristic function of the weighted sum is the product of characteristic functions of these distributions:  $\psi_{w_xx+w_yy}(t)=\psi_x(w_xt)\times \psi_y(w_yt)$. You can obtain the distribution $F_{w_xx+w_yy}$ from its characteristic function, of course.
Finally, the whole idea of the Gaussian mixture is to model unknown distributions, such as those with fat tails in finance.
Now, what is meant by $w_x\mathcal N+w_y\mathcal N$? In case of a mixture it is the weighted average density (PDF).
