Past exam of the probability course of the University of Oxford / 2015 / 3 / b / ii / Solution

For $k=0$, the target is reached at time zero, so $h_0=1$. For $k<0$, the preceding almost sure convergence gives $Y_n\to +\infty$ with probability one. A sample path with unit upward and downward steps must visit every intermediate integer when it crosses from a negative value to a positive value. It must therefore visit zero at a finite time. ThusThe unit-step property matters: positive long-run displacement alone would not force a walk with larger upward jumps to land exactly on zero.