Let $(X_n)$ be an irreducible, aperiodic, positive recurrent, time-homogeneous DTMC on the integers. Let $\pi$ denote the limiting/stationary distribution. It is known that the expected deviation from $0$ associated with the limiting distribution is finite; that is, $\sum_{i \in \mathbb{Z}} |i| \pi_i = L < \infty$.
If I fix the initial state, for example $X_0 = 0$, can I say anything about the limit of the sequence $(\Bbb{E}[X_1 | X_0 = 0], \Bbb{E}[X_2 | X_0 = 0], \Bbb{E}[X_3 | X_0 = 0], \ldots)$?
Intuitively (and perhaps incorrectly), I would like to say that $\lim_{n \to \infty} \Bbb{E}[X_n | X_0 = 0] = L$. Unfortunately, probability is not my area of research, so I am not sure how to prove it, or if this is a known result, or if additional conditions are required.
Upon further searching various rephrasings of the problem, the limit is indeed $L$ as proved in this recent Math StackExchange question's accepted answer.