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Consider a symmetric random walk on $\mathbb{Z}/m \mathbb{Z},$ i.e. we start in some state $[k]$ and then propagate with equal rates either to $[k+1]$ or $[k-1]$ and so on. How do I show that this process is ergodic. By ergodic I mean

$$E^{[k]}(f(X_t))\rightarrow \int f(X_t(\omega))dP(\omega)$$ for some measure $P$ where $f \in C_b$ and $X_t$ is the process. $E^k$ is the expected value of the process started with respect to the measure $d \delta_{[k]},$ i.e. almost surely in state $[k]$.

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closed as off-topic by Math1000, Claude Leibovici, Silvia Ghinassi, jameselmore, Did Jan 28 '16 at 14:51

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