When does the screening effect not hold?

The screening effect is the phenomenon of nearby observations yielding a good approximation to the optimal linear predictor of a spatial process based on a large set of observations. In addition to its obvious relevance to computation with large spatial datasets, knowing when a screening effect occurs is key to understanding the behavior of spatial processes. This work provides the first general results showing when an asymptotic screening effect does not hold by considering the prediction of a weakly stationary, isotropic process on Rd at a single location based on observing the process everywhere on Rd with white noise and letting the variance of the white noise tend to 0. The main result shows that a screening effect does not hold if the isotropic spectral density fluctuates too much at high frequencies.