TITLE: Does ergodicity imply stationarity?
QUESTION [6 upvotes]: In general, if a random process is ergodic, does it imply that it is also stationary in any sense?

REPLY [2 votes]: In the theory of Dynamical Systems, one usually only defines ergodicity for invariant measures, so the short answer could be 'yes'.
But, consider the following theorem (taken from Norris' Markov Chains):
Theorem 1.10.2: let $P=(p_{ij}:\,i,j\in I)$ be an irreducible, positive recurrent transition matrix indexed by $I\times I$, where $I$ is a countable set, and let $\lambda = (\lambda_i:\,i\in I)$ be any distribution on $I$. If $(X_n:\,n\in\mathbb{Z}_+)$ is a Markov chain with initial distribution $\lambda$ and transition matrix $P$, then for any bounded function $f\colon I\to \mathbb{R}$ there is a set $\Omega_f$ with $\mathbb{P}(\Omega_f) = 1$ such that $$\frac{1}{n}\sum_{k=0}^{n-1} f\circ X_k(\omega) \to \sum_{i\in I} \pi_i f(i),\qquad \forall\omega\in \Omega_f,$$ where $\pi = (\pi_i:\,i\in I)$ is the unique invariant distribution of $P$.
By taking $\lambda\neq \pi$, clearly the corresponding chain is not stationary, but nevertheless the time averages converge to the "limiting space averages".