Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 2 / 18H / b

The chain is a birth-death chain. Detailed balance between $0$ and $1$ givesFor $i\ge 1$, detailed balance between $i$ and $i+1$ givesConsequentlyThe expected occupation time of the positive even states during a return cycle to state $1$ is therefore

Normalizing the invariant distribution found in part (i) givesThe mean recurrence time formula now yields

Let $h_i={\mathbb{E}}_i{T}_0$ be the expected hitting time of state $0$. For $i\ge 1$, first-step analysis givesWith $d_i=h_i-h_{i-1}$ this becomesThe minimal nonnegative solution hasso

Equivalently, Kac's lemma gives ${\mathbb{E}}_0{T}_0^{+}=1/{\pi }_0$. A first step from $0$ either returns immediately with probability $1-p$ or moves to $1$ with probability $p$, soSubstituting ${\pi }_0=(q-1)/(q-1+q^{2}p)$ gives the same result.