Limiting Stochastic Operations for Stationary Spatial Processes

A natural extrapolation of stochastic operations (continuity and differentiation) already described in time domain (one-dimensional case) is established for spatial processes (two- or three-dimensional case). If stationarity decision is assumed, the continuity and differentiability (in the mean square sense) of a spatial process depends on the continuity and differentiability of the correlation function at the origin. Spatial processes described by stationary random functions are not continuous (in the mean square sense) when the covariance function presents a nugget effect, and they are not differentiable when the same covariance function is described by a spherical or an exponential covariance.