Are Gaussian mixtures stable?

Question

Let $X$ be a Gaussian-mixture random vector with probability density function (PDF) $f_X(x)=\sum_{i=1}^kp_if_i(x)$, where for $i=1,2,\ldots,k$, $f_i$ is a multivariate Gaussian PDF with mean $\mu_i$ and covariance matrix $\Sigma_i$, and $\sum_{i=1}^kp_i=1$. Is $Y=AX$, where $A$ is an arbitrary matrix of appropriate size, also a Gaussian-mixture random vector? If yes, what are the parameters of its PDF?

Answer 1

This is still a mixing of Gaussian variates $N(A^T\mu_i, A^T\Sigma_i A )$ with the same mixing coefficients $p_i.$ Proof: compute the Laplace transform $AX.$