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

$\boxed{(b)⟷P3}$. 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.