Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 18H / a / Solution

The chain is an irreducible birth-death process. The detailed balance equations areso ${\pi }_{i+1}={\pi }_i/2$. Normalization gives the stationary distributionAn irreducible countable-state Markov chain that possesses a stationary probability distribution is positive recurrent. Hence $X$ is positive recurrent.