# Gaussian ARMA process with non-gaussian innovations

Can a gaussian ARMA process have non-gaussian innovations? (i.e., is there an ARMA process that is gaussian, but the corresponding innovations are not gaussian)?

-
No............. –  Rob Hyndman Mar 9 '12 at 1:30
many thanks for your comment. but could you provide a sort of explanation for your answer (or a hint to literature that provides a formal proof or something the like?) –  s_2 Mar 10 '12 at 3:19
@mpiktas has provided the details in his answer. –  Rob Hyndman Mar 10 '12 at 4:07

If a process is Gaussian, then any linear combination of its elements is Gaussian too. Take for example the AR(1) process:

$$X_t=\phi X_{t-1}+\varepsilon_t$$

Then

$$\varepsilon_t=X_t-\phi X_{t-1},$$

i.e. the innovation is a linear combination of the process elements, hence it should be Gaussian if $\{X_t\}$ is Gaussian.

-
@Rob, thanks for edits. The articles in English will remain a mystery for me :) –  mpiktas Mar 9 '12 at 7:11
thanks for your comment. i thought something along the same lines... but does it hold for a general ARMA(p,q) process as well? if so, how would you prove the statement formally (w/o assuming invertibility, causality etc.) –  s_2 Mar 10 '12 at 3:21
what if there is an MA part too? –  s_2 Mar 10 '12 at 14:10