Suppose $X_n$ is an irreducible markov chain with transition probability matrix $P_{ij}$. My text books says that if there exists $\pi_i>0$ for all $i$ such that $\pi_iP_{ij} = \pi_jP_{ji}$ for all $i,j$, then $\pi_i$ are the stationary probabilities of $X_n$.
My question how can I prove that
- $X_n$ is recurrent, i.e. that starting from state $i$, the probability of returning to $i$ is one?
- Is it possible to see that the conditions above imply that the markov chain is ergodic?
Define $\tau^i_k := \inf \{n\geq \tau_{k-1}^i : X_n = i\}$, $\tau_0 := 0$. Assume $(X_n)_n$ is transient, then $\Bbb P _j(\tau^i = \infty) > 0$.
Note that with the Markov Property we have $$\sum_{n=0}^\infty (P^n)_{ji} = \sum_{n=0}^\infty \Bbb P _j(X_n = i) = \Bbb E _j\left[\sum_{n=0}^\infty 1_{\{X_n = i\}}\right]\\ = \sum_{k=1}^\infty k \Bbb P_j (\tau_k^i < \infty, \tau_{k+1}^i = \infty) = \Bbb P_j(\tau_1^i <\infty)P_i(\tau_1^i =\infty)\sum_{k=1}^\infty k \Bbb P_i (\tau_1^i < \infty)^k< \infty$$ which implies $$(P^n)_{ji} \rightarrow 0$$ as $n \rightarrow\infty$. If $\pi$ is a non trivial invariant probability measure, there is an $i$ with $\pi_i > 0$, but $\pi_i = (\pi P^n)_i \rightarrow 0$. Contradiction.
I have no answer to your question about ergodicity yet, but it seems to me like a coupling argument.