# Convergence of means of an ergodic Markov chain

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 ?

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 Many ergodic irreducible Markov chains do not converge in distribution. You might indicate whether you assume that the chain converges in distribution. – Did Sep 12 '12 at 10:10