Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 219 / 3 / i

$\boxed{(a)⟷P2}$. The exponential covariance function gives a stationary Ornstein-Uhlenbeck process. At a small lag $h$,The variance of a difference quotient therefore diverges as $h\to 0$, so the process has no mean-square derivative. Its continuous version has rough, nondifferentiable paths, matching the jagged P2 trace in the original PDF.

$\boxed{(b)⟷P3}$. This is a Gaussian kernel covariance. Near zero, $2A^2(1-e^{-h^2/{\tau }^2})\sim 2A^2{h}^2/{\tau }^2$, consistent with a finite mean-square derivative variance. The kernel has derivatives of every order, and the resulting Gaussian process admits a smooth version. P3 is smooth but does not repeat, unlike P1.

$\boxed{(c)⟷P1}$. A periodic covariance function gives identical latent values one period apart: if $p$ is its period, $\mathrm{Var}(f(t+p)-f(t))=k(t+p,t+p)+k(t,t)-2k(t+p,t)=0$. The smooth repeating P1 trace is therefore the appropriate match.

The original PDF has ${\ell }^2$ in the prefactor, not the TeX transcription's $t^{\prime 2}$. It prints ${\sin }^2(2\pi |t-t^{\prime }|/T)$; this kernel repeats at $\boxed{p=T/2}$ because ${\sin }^2(x+\pi )={\sin }^{2}x$. To parameterize the period itself by $P$, use ${\sin }^2(\pi (t-t^{\prime })/P)$ as below. The absolute value is immaterial inside the squared sine.