When predicting the value of a stationary random field at a location
x in some region in which one has a large number of observations, it may
be difficult to compute the optimal predictor. One simple way to reduce the
computational burden is to base the predictor only on those observations
nearest to x. As long as the number of observations used in the
predictor is sufficiently large, one might generally expect the best predictor
based on these observations to be nearly optimal relative to the best predictor
using all observations. Indeed, this phenomenon has been empirically observed
in numerous circumstances and is known as the screening effect in the
geostatistical literature. For linear predictors, when observations are on a
regular grid, this work proves that there generally is a screening effect as
the grid becomes increasingly dense. This result requires that, at high
frequencies, the spectral density of the random field not decay faster than
algebraically and not vary too quickly. Examples demonstrate that there may be
no screening effect if these conditions on the spectral density are
violated.