Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 217 / 4 / Solution

Let $e_n(t)=\sqrt{2}\sin (n\pi t)$, let $(g_n)$ be independent standard normal variables, and choose $a_n=e^{-n^2}$. DefineFinite collections of values are limits of centered Gaussian vectors, so this is a centered Gaussian process.

The functions $(e_n)$ are an orthonormal basis of $H=L^2(0,1)$. Hencewhich proves ${\mu }_X(H)=1$. For every integer $k\ge 0$,Thus, almost surely and simultaneously for all $k$, the differentiated series converges uniformly on $(0,1)$. Termwise differentiation gives an almost surely infinitely differentiable version.

Finally, let $F\subset H$ be finite-dimensional. Choose a nonzero $h\in F^{\perp }$. Thenis a centered normal variable of variancebecause every $a_n$ is positive and $(e_n)$ is complete. The event $\{X\in F\}$ is contained in $\{⟨X,h{⟩}_H=0\}$, which has probability zero because a nondegenerate normal distribution has no atoms. Therefore ${\mu }_X(F)=0$ for every finite-dimensional linear subspace $F$.

Solved by gpt-5.6-sol high.