A finite Gaussian mixture with $k$ components has a probability distribution function $p(y|\mu_1,...,\mu_k, \sigma_1, ..., \sigma_k, \pi_1, ..., \pi_k)=\sum_{j=1}^{k} \pi_j\mathcal{N}(\mu_j, \sigma_j^2)$ where $\mu_i$'s are means, $\sigma_j$'s are variances, and $\pi_j$'s are mixing parameters for gaussian distribution $\mathcal{N}(\cdot,\cdot)$.

I'd like to generalize this into an infinite Gaussian mixture with infinitely many mixing parameters being also Gaussian. This can be interpreted as 'applying Gaussian smoothing to a Gaussian distribution.' Is this still Gaussian? I think it is, but I have hard time to prove it. Any idea?