 Solution

A Discrete-time Markov chain with initial distribution $\hat{\pi }$ is reversible if $(Y_0,\dots ,Y_n)$ and $(Y_n,\dots ,Y_0)$ have the same law for every $n$. Its detailed balance equations are ${\hat{\pi }}_i{\hat{P}}_{ij}={\hat{\pi }}_j{\hat{P}}_{ji}$. These equations are equivalent to reversibility. Indeed, reversing every factor in the probability ${\hat{\pi }}_{i_0}{\prod }_{r=1}^n{\hat{P}}_{i_{r-1}{i}_r}$ gives the reverse path probability; the converse follows by taking $n=1$.

Summing detailed balance over $i$ givesHence $\hat{\pi }$ is an stationary distribution.