Consider two independent random variables $X, Y$ which are two Gaussian mixture model (GMM) in 1-dim space.
Let $Z = f(X, Y)$, is R.V. $Z$ still a GMM?
I randomly generated some GMMs and function $f(\cdot, \cdot)$ and it seems that the distribution of $Z$ is still GMM but I am not sure how to prove this (and also I am not sure the conclusion can be drawed).
I appreciate a lot for any hint.
I don't think you can possibly prove this without further restrictions on $f$. For instance, suppose $f(x, y) = 0$. Then $Z$ is deterministic. Likewise you can easily design $f$ to be piecewise such that $Z$ is distributed as a Bernoulli variable with some parameter depending on $X,Y$. So in general $Z$ won't be distributed as a mixture of gaussians.