Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 19H / b / Solution

There are no visits when the initial excursion returns to $i$ before hitting $j$, soFor $k\ge 1$, the chain must first hit $j$, return to $j$ before $i$ exactly $k-1$ times, and then hit $i$ before another return to $j$. The Strong Markov property givesThe formula also covers the boundary cases $\alpha =0$ and $\alpha =1$.