Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 201 / 2 / d

Write $p_m=\mathbb{P}(S_T=m)$, $p_{m+1}=\mathbb{P}(S_T=m+1)$, and $q_m=p_m+p_{m+1}$. Part (b) givesso $q_m\le 2^{-m}$. The exit-state decomposition also givesCombining this with part (c), including the upper overshoot, yieldsFor example, the same calculation supplies the quantitative boundThe asymptotic linear growth is determined by the negative mean increment; the exponentially unlikely upper exit gives a vanishing correction.