Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 1 / 19H / c / Solution

In the positive-recurrent case, let $q=p_0+p_1<p$. Detailed balance for the length chain requiresso, with $r=q/p<1$,Normalization gives ${\pi }_0=1-r$. The self-loop at zero makes the length chain aperiodic, so the convergence theorem for irreducible positive-recurrent aperiodic Markov chains gives