I have the following pdf
$\sum_{A=0}^{\infty}\cfrac{\rm{e}^{\lambda_S}[\lambda_S]^A}{A!}\int_{S^{A}}\phi_n(\mu,\Sigma)\rm{d}S_N$,
where $S$ is some area (we can imagine a 10x10 rectangle), $S_N \in \mathbb{R}^2$ are the locations i.e. if $A=1$ we have $1$ location, if $A=2$ we have $2$ location..., $\lambda_S \in \mathbb{R}^+$, $A\in \mathbb{N}$, $\phi_n$ is a $n$-variate Normal pdf, $\mu\in\mathbb{R}^n$ and $\Sigma$ is a isotropic covariance function depending only on the locations $S_N$.
I am struggling to show that this "mixture" can never be a Gaussian distribution. Any hint?