Choose step functions $f_n\to f$ in $L^2[0,1]$. Part (i) shows that $\int B_sf_n(s)ds$ is centered Gaussian. Since the kernel $K(r,s)=\min(r,s)$ is bounded,
$$
\mathbb E\left|\int_0^1B_s(f_n-f)(s)ds\right|^2
=\iint(f_n-f)(r)(f_n-f)(s)K(r,s)drds\to0.
$$
The $L^2$ limit is therefore centered Gaussian, and its variance is
$$
\int_0^1\int_0^1f(r)f(s)\min(r,s)\,dr\,ds.
$$


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-201/1/b/ii/solution