Will state space model be affected by Gaussian mixture shock

Dear All,
I just have a question. If one of my structural shocks follows a Gaussian mixture process, and I included it into my DSGE model, will my DSGE model can still be transformed into a state space model after log linearization?
Thank you and any ideas will be helpful!

Due to the way first order perturbation works, your shock will always be approximated via its mean and variance, effectively being rendered a normal distribution.