It's "well known" that when a Gaussian random process is input to a linear system, the output is also a Gaussian random process.
This video formally proves this result for a stable discrete-time linear system with a 2nd order Gaussian random process input. The key steps in this proof are:
1) Since the system is stable and the input is a 2nd order process, the output random process Y(k) is well-defined, and for any arbitrary time in the limit is equal in a mean-square sense to a finite sum of Gaussian random variables. We denote this finite summation of Gaussian random variables as YN(k).
2) Since the output random process is MS convergent to YN(k), it is also convergent in distribution. This allows us to solve for the distribution of Y(k) by taking the limit of the distribution of YN(k). By taking this limit we find that Y(k) has a Gaussian distribution as desired.