= 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 is
$$
c_0=\frac1P\int_0^Pk_P(t,u)\,du=A^2e^{-\ell^{-2}}I_0(\ell^{-2}),
$$
where $I_0$ is a <modified Bessel function>. Replacing $f(t)$ by $f(t)-P^{-1}\int_0^Pf(u)\,du$ gives a Gaussian process with kernel $k_P(t,t')-c_0$. 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.
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