Suppose $\{X_n\}$ is an ergodic Markov Chain with general state space $\mathcal{X}$ and stationary distribution $\pi$. Suppose $\forall x\in\mathcal{X}, f(x)\geq 0$ and $\mathbb{E}_{\pi}[f(X)]<\infty$. Is it possible to show, maybe with additional assumptions, that the sequence of random variables $\{f(X_n)\}$ is uniformly integrable?
The point is, whether ergodicity of $\{X_n\}$ can play a role in showing uniform integrability of $\{f(X_n)\}$ without assuming boundedness of the latter?
I have a vague intuition but I'm not sure whether it is on the right track: Since $f(X_{\infty})$ is integrable, then maybe, if $f(X_1)$ is integrable (an assumption on the initial distribution really), then everything between $f(X_1)$ and $f(X_\infty)$ is integrable because the transitions are "ergodic".
I'm not sure if my intuition makes sense. Any help is highly appreciated!