Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 326 / 4

A Gaussian measure $\mu$ on a real separable Banach space $X$ is a Borel probability measure such that ${\ell }_{\ast }\mu$ is a one-dimensional normal distribution for every continuous linear functional $\ell \in X^{\ast }$. Its mean $m\in X$ and covariance operator of a Gaussian measure $C:X^{\ast }\to X$ are characterized byandfor all ${\ell }_1,{\ell }_2\in X^{\ast }$.

The indicator functions satisfyTherefore ${\phi }_1,{\phi }_2\in L^2[0,1]$.

For every $h\in L^2[0,1]$, the continuous linear functional induced by the inner product givesThis is a normal random variable because it is a linear combination of independent normal random variables. Hence the law of $U$ is a Gaussian measure. Its mean is zero, and independence together with $\mathrm{Var}({\xi }_i)=1/2$ givesThus its covariance operator of a Gaussian measure is

The supports of ${\phi }_1$ and ${\phi }_2$ are disjoint, so they are orthogonal vectors. Substitution in the covariance formula givesandTherefore the associated eigenvalues are

Put $c_i={\int }_D{\phi }_i(x)d{\lambda }_n(x)$. The stated scalar random variable isAs a finite linear combination of independent normal random variables, it is normally distributed. Its mean is zero and its variance is

Finally, orthonormality of the ${\phi }_i$ givesSince $\mathbb{E}{\xi }_i^2=1/2$,