Does the marginals of mixtures of Gaussians follow the properties of Gaussian distribution and the definition of marginalization? What I want to do is to obtain the marginal probability density function of each $x_j$, for all $j=1,\ldots,n$, where $X=(x_1,\ldots,x_j,\ldots,x_n)^\top$ in a Gaussian mixture model $p(x)=\Sigma_{k=1}^Kw_kN(x|\mu_k,\Sigma_k)$
For the Gaussian distribution I know from the Wikipedia's article to find the marginal distribution I can simple to this.: (And I've seen a lot of proof for this.)
To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix.
I have seen that the marginals of Gaussian mixtures will also be Gaussian mixtures, but I haven't come across any mathematical proof to support this. Does this mean that, as I will do with the Gaussian distribution, I can just use the diagonal entries of the Sigma (of each component) for the variances of the random variables? And is there a proof for this?