  In this work, we investigate Gaussian process regression used to recover a
function based on noisy observations. We derive upper and lower error bounds
for Gaussian process regression with possibly misspecified correlation
functions. The optimal convergence rate can be attained even if the smoothness
of the imposed correlation function exceeds that of the true correlation
function and the sampling scheme is quasi-uniform. As byproducts, we also
obtain convergence rates of kernel ridge regression with misspecified kernel
function, where the underlying truth is a deterministic function. The
convergence rates of Gaussian process regression and kernel ridge regression
are closely connected, which is aligned with the relationship between sample
paths of Gaussian process and the corresponding reproducing kernel Hilbert
space.
