Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 219 / 2 / a / Solution

The squared chord distance isHence the restricted Gaussian process has covarianceIt depends only on $t-t^{\prime }$, so it is stationary. It is periodic in either argument with period $T=2\pi /\omega$, and Gaussian-process realizations inherit that period almost surely because $f(t+T)-f(t)$ has zero variance.