27 - Detailed Balance Implies Stationary

Detailed Balance Implies Stationary

Let $P$ be a transition matrix with distribution $\pi$ then

If $P$ and $\pi$ are in detailed balance then

$$\pi \text{ is stationary for }P$$

Proof

For any $j$ then

$${\pi }_j={\pi }_j\sum _i{p}_{ji}=\sum _i{\pi }_j{p}_{ji}=\sum _i{\pi }_i{p}_{ij}$$