  The posterior variance of Gaussian processes is a valuable measure of the
learning error which is exploited in various applications such as safe
reinforcement learning and control design. However, suitable analysis of the
posterior variance which captures its behavior for finite and infinite number
of training data is missing. This paper derives a novel bound for the posterior
variance function which requires only local information because it depends only
on the number of training samples in the proximity of a considered test point.
Furthermore, we prove sufficient conditions which ensure the convergence of the
posterior variance to zero. Finally, we demonstrate that the extension of our
bound to an average learning bound outperforms existing approaches.
