Periodic covariance function ($k_P(t,t^{\prime })=A^2\exp [-2{\ell }^{-2}{\sin }^2(\pi (t-t^{\prime })/P)]$)

This covariance function is periodic with period $P>0$. Map time to the circle $u(t)=(\cos (2\pi t/P),\sin (2\pi t/P))$ and pull back a Gaussian kernel; since $\Vert u(t)-u(t^{\prime }){\Vert }^2=4{\sin }^2(\pi (t-t^{\prime })/P)$, this proves positive semidefiniteness. A Gaussian process with this kernel has $\mathrm{Var}(f(t+P)-f(t))=0$ and therefore repeats at every fixed phase. Replacing the inner $\pi$ by $2\pi$ changes the actual period to $P/2$.

A zero-mean Gaussian process need not have zero average in each realization. For the periodic covariance function, the constant Fourier component iswhere $I_0$ is a modified Bessel function. Replacing $f(t)$ by $f(t)-P^{-1}{\int }_0^{P}f(u)du$ gives a Gaussian process with kernel $k_P(t,t^{\prime })-c_0$. This is a positive-semidefinite kernel because it is a covariance after a linear transformation, and each sample has zero period average. The real integral representation of the modified Bessel function I0 yields the displayed constant.