Gaussian approximations to conditional distributions for multi-Gaussian processes

Suppose a multi-Gaussian process is observed at some set of sites, and we wish to obtain the conditional block grade distribution given some observations. We show that this conditional distribution is approximately Gaussian under certain conditions. In particular, given a single observation from a continuous multi-Gaussian process, the conditional distribution under a small change of support is approximately Gaussian unless, roughly speaking, the observed process is twice differentiable and the observation site is at the center of mass of the support region. A Gaussian approximation for the conditional prediction error of the total ore in a fixed region is considered also, although an example demonstrates that a naive analysis can give incorrect limiting conditional means.