Past exam of the probability course of the University of Oxford / 2023 / 3 / b / iii / Solution

Assume $p<1/2$, and put $a=\theta /p$ and $r=p/q\in (0,1)$. The detailed balance equations giveMultiplying from $0$ to $k-1$, the last factor telescopes, while the product involving $a$ is evaluated by the Gamma function:Consequently the requested invariant distribution can be expressed using the paper's power series asThe assumed convergence of $g_{p,\theta }(r)$ ensures that this normalization is finite and positive.

For an explicit normalizing constant, the binomial series givesThe termwise differentiation of a power series is valid for $|r|<1$, soThe last equality uses $r=p/q$ and $a=\theta /p$: $ar+\theta (1-r)=\theta /q+\theta (q-p)/q=2\theta$. ThereforeThese positive terms sum to one and satisfy the detailed balance equations, so part (a)(ii) verifies stationarity. Irreducibility makes the invariant distribution unique.