Example of a Markov Chain which is stationary but not reversible

Question

There are a few questions on this site which ask for an example of a stationary but not a reversible Markov Chain. However most of them go straight to talking about irreducibility. Is there an explicit example which directly shows how reversibility is different to stationarity?

Answer 1

The simplest example of a transition matrix giving a non-reversible Markov chain is something like $$ \begin{pmatrix} 1/2 & 1/2 & 0 \\ 0 & 1/2 & 1/2 \\ 1/2 & 0 & 1/2 \end{pmatrix} $$ You can think of a particle moving around a triangle. It can stay where it is, or move clockwise, but it can never move anticlockwise. If you take a video and play it backwards, you will see different behaviour - the particle now moves anticlockwise around the triangle, and never clockwise. So the process is not time-reversible.

In algebraic terms, you can see that the detailed balance conditions don't hold. The stationary distribution is uniform: $\pi=(1/3, 1/3, 1/3)$. You can check for example that $\pi_1 p_{12} = 1/3\times1/2 = 1/6 \ne 0 = \pi_2 p_{21}$.