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 $[k]$.


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

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Math1000, Claude Leibovici, Silvia Ghinassi, jameselmore, Did
If this question can be reworded to fit the rules in the help center, please edit the question.