Question involving an invariant measure on a Markov chain

Question

Suppose $\mu$ is an invariant measure for a Markov chain with state space $S$ with $\mu(i)p_{ij}=\mu(j)p_{ji}$ $\forall i,j \in S$. Describe a Markov chain with this property. Also, show that $\mu$ is an invariant measure.

In constructing the Markov chain, I know I want it to have transition matrix $P$ that satisfies $\mu'=\mu'P$ (prime denoting transpose). I can't think of a named markov chain off the top of my head with this property. Perhaps Gambler's Ruin or deterministically monotone?

To prove it's invariant measure, I need $\sum\limits_{i \in S} \mu(i) \neq 1$. Don't I need the chain before I can do this?

Answer 1

• an example is the transition matrix$\bigl(\begin{smallmatrix} 0&1\\ 1&0 \end{smallmatrix} \bigr)$ with $\mu_1 = \mu_2 = \frac 12$.
• If $\mu_i p(i,j) = \mu_j p(j,i) $ then summing over $i$ puts $$ \mu_i= \sum_j \mu_i p(i,j) = \sum_j \mu_j p(j,i) = (\mu p)_i $$that is: $\mu$ is invariant.