Let $X_{n}$ be an ergodic Markov chain evolving on $\mathbb{Z}_+$. Let $\pi$ be the invariant distribution. Is it true that $\lim_{n \rightarrow \infty} \mathbb{E} X_{n} = \mathbb{E} X $ ? Are there any conditions under which such a limit holds ?
I was thinking along the following lines : it is known that $X_{n} \rightarrow X \sim \pi$ in distribution. So if $X_{n}$ is uniformly integrable then $\mathbb{E}X_{n} \rightarrow X$ ? But is uniform integrability required ?
