I am trying to solve the following:
Let $P$ be a matrix of transition probabilities of a homogeneous ergodic Markov chain on a finite state space such that $p_{ij} = p_{ji}$. Find its stationary distribution.
I know from the Ergodic Theorem for Markov Chains that since the Markov chain is ergodic there exists a unique stationary distribution $\pi$ and the n-step transition probabilities converge to the distribution $\pi$ that is $\lim_{n \to \infty} p_{ij}^{n}=\pi_{j}.$
How do I actually apply the Ergodic Theorem for Markov chains to compute the stationary distribution in this case.
Because $p_{ij} = p_{ji}$ for all $i,j$, we may state that $p_{ij}^n = p_{ji}^n$ for any $n$ (and any $i,j$). It therefore follows that for all $i,j$, we have $$ \pi_i = \lim_{n \to \infty} p_{ji} = \lim_{n \to \infty} p_{ij} = \pi_j $$ conclude that $\pi$ is the uniform distribution.