Periodic Gaussian process with zero period average

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.