$\epsilon_{t}$ is a stationary AR(1) process. Hence it is also ergodic.
What can be said of $\mu_{t}=\log(\epsilon_{t}+1)$? Is such a process well defined? What would be the conditions on the AR(1) process $\epsilon_{t}$ for the $\mu_{t}$ process to be well defined?
Would $\mu_{t}$ also be stationary and ergodic?
I would like to use some kind of central limit argument (CLT for non-i.i.d. data) to determine the distribution of $\sum_{t=0}^{n}\mu_{t}$, but am not sure I can.