Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 1 / 20H / c / Solution

Let the common initial state be $x$ and the robber's final state be $y$. Part (a) gives their joint distribution asLet ${\tau }_y=\min \{n\ge 0:X_n=y\}$ be the cop's hitting time of $y$ when starting from $x$. For a chain started from $y$, condition on its first step to obtainHence the expected catch time, averaged over the displayed joint law, isUsing Kac's lemma once more givesThe convention ${\tau }_y=0$ correctly counts an immediate catch when the robber's reverse step leaves it at the common starting state.