This is more of a "ask-for-idea" than "ask-for-answer" question:
Suppose $\{X_t\}$ is an ergodic process with a known stationary\limiting distribution $\pi$. Let $f(t)$ be a deterministic and increasing function. Is the time-changed process $Y_t:=X_{f(t)}$ ergodic? If it is, whether its stationary\limiting distribution remains as $\pi$?
Intuitively the answers to both of the above questions are yes. Any relevant reference, counter-example or just general insights in how to solve the above questions are highly appreciated!
Thanks in advance!