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

Writing $c_i(t)=⟨1_{[0,t]},f_i{⟩}_{L^2}$ gives $X_n(t)={\sum }_{i=1}^n{\alpha }_i{c}_i(t)$, henceBy Parseval identity, the variance tends to $\Vert 1_{[0,t]}{\Vert }_2^2=t$. Thus $X_n(t)\stackrel{d}{\to }N(0,t)$.

For fixed $t$, $(X_n(t))$ is a martingale with independent centered increments andThe $L^2$ Martingale convergence theorem gives convergence both almost surely and in $L^2$ to a random variable $X(t)$.

For real $u_1,\dots ,u_j$, the preceding series construction givesIt is centered Gaussian, and Parseval identity makes its varianceThis is the variance of ${\sum }_r{u}_r{B}_{t_r}$. The Cramer-Wold theorem proves equality of the two joint distributions.