An ARMA(p,q) process is a (weakly) stationary process $x_t=\sum_{i=1}^p\phi_ix_{t-i}+z_t+\sum_{j=1}^q\theta_j z_{t-j}$ where $z_t$ is white noise.
Lets assume that $z_t$ is Gaussian white noise. That means $z_t$ is independent (Gaussian + uncorrelated $\rightarrow$ independent) and normal distributed with mean zero and finite variance $\sigma^2>0$.
I was told, that then $x_t$ is also Gaussian. But I do not know how to prove that.