Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 201 / 1 / c / Solution

Use the natural filtration ${\mathcal{F}}_k=\sigma (X_1,\dots ,X_k)$ and define $M_k=S_k-mk$. The increments $X_k-m$ are centered, independent, and square integrable, so their conditional expectations given the past are zero. Thus $(M_k)$ is a square-integrable martingale with $M_0=0$. Independence givesApplying the Doob L2 maximal inequality yieldsThis includes ${\sigma }^2=0$, when every centered increment vanishes almost surely.