Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 2 / 12F / i / Solution

For $x\ge m$, $X_n=x$ implies $M_n\ge m$. For $x<m$, reflect every step after the first visit to $m$. The reflection principle for simple symmetric random walk bijects such paths ending at $x$ with unrestricted paths ending at $2m-x$. Therefore