Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 6 / b / Solution

Write $\mathbb{E}{X}_1=\mu$ and $\mathrm{Var}(X_1)={\sigma }^2<\infty$. For rational $t=m/n$, stationarity and independence of the $n$ increments over intervals of length $1/n$ giveusing variance additivity for independent random variables. Stochastic continuity extends both identities from rational to real $t$. In the centered case $\mu =0$, this becomes $\mathbb{E}{X}_t=0$ and $\mathbb{E}{X}_t^2=t{\sigma }^2$.

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