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

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