Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 1 / b

For a partition of $[a,b]$, every Riemann sum $S_n={\sum }_j{B}_{t_j}\Delta t_j$ is a centered Gaussian random variable, withPath continuity gives $S_n\to {\int }_a^b{B}_{s}ds$ almost surely, and the covariance bound $\mathbb{E}|B_s-B_t{|}^2=|s-t|$ also gives convergence in $L^2$. Hence the limit is Gaussian, centered, and passage to the limit in the displayed sums gives variance

Choose step functions $f_n\to f$ in $L^2[0,1]$. Part (i) shows that $\int B_s{f}_n(s)ds$ is centered Gaussian. Since the kernel $K(r,s)=\min (r,s)$ is bounded,The $L^2$ limit is therefore centered Gaussian, and its variance is