Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 3 / 9H / a / Solution

A reversible Markov chain has a stationary law $\pi$ for which every finite path has the same distribution as its time-reversed path:For a transition matrix $P$, the equivalent detailed balance condition is ${\pi }_i{P}_{ij}={\pi }_j{P}_{ji}$ for all $i,j$. It implies $\pi P=\pi$ by summing over $i$. Multiplying the balance identities along a path shows equality of its forward and reverse probabilities when $X_0\sim \pi$; conversely, taking paths of length one gives those identities.