Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 7H / a / Solution

Let $q={\mathbb{P}}_i(T_i<\infty )$. By the Strong Markov property, after every return to $i$ the chance of another return is again $q$. HenceThus $V_i$ is geometric with success parameter $1-q$; when $q=1$, it is infinite almost surely.