Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 1 / c / Solution

Suppose $c>1$ and $\mathbb{E}[T_c]<\infty$. Part (a) would giveBut $T_c\ge 1$ and its defining strict inequality gives $S_{T_c}^2>c^2{T}_c$ almost surely. Taking expectations would yieldwhich is impossible because $c^2>1$. Part (b) shows that $T_c$ is nevertheless finite almost surely. Consequently