Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 33 / 4 / c / Solution

The strong law of large numbers gives $S_n/n\to -1$ almost surely, so $S_n\to -\infty$. Therefore the infinite-horizon maximum $Z$ is finite and attained almost surely; also $Z\ge 0$ because $S_0=0$.

Apply the Doob maximal inequality to $M_k=e^{2S_k}$ through time $n$. Since $\mathbb{E}{M}_n=1$,These finite-horizon events increase to $\{Z>t\}$. Continuity of probability under increasing unions givesFor $t<0$ the same bound is trivial, because its right side exceeds $1$. Only bounded-horizon maximal inequalities were used before passing to the limit; no unjustified stopping at an infinite-horizon crossing time is needed.