Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 225 / 1

Write $Z=X-\mu$, so the covariance operator isFor $g,h\in L^2[0,1]$,so $C_X$ is self-adjoint. Moreover,so it is a positive operator.

Let $(e_j)$ be any orthonormal basis. Tonelli theorem and Parseval identity giveA positive operator with finite trace is a trace-class operator, completing the proof.

For every orthonormal basis $(e_j)$ of the infinite-dimensional Hilbert space $L^2[0,1]$, the identity operator satisfiesPart a shows that every covariance operator of a square-integrable Hilbert-space random element is trace-class. Therefore the identity cannot be a covariance operator.

Choose a unit vector $e$ and define the rank-one operatorIt is bounded, self-adjoint, and Hilbert-Schmidt, with $\Vert T{\Vert }_{\mathrm{HS}}=1$. Butso it is not positive. Since every covariance operator is positive, $T$ is the required counterexample.

Linearity of the Bochner integral givesIndependence and centering of $X_i-\mu$ imply

Since $C_X{\varphi }_k={\lambda }_k{\varphi }_k$ and the eigenfunctions can be chosen orthonormally, independence of the sample givesThis remains valid for repeated eigenvalues after choosing an orthonormal eigenbasis within each eigenspace.