Screening effects in Gaussian random fields under generalized spectral conditions

The screening effect is a central idea in spatial prediction: once nearby observations are used, distant ones add little. While Stein’s classical results explain this effect under strong spectral conditions, many models used in practice fall outside those assumptions. In this paper, we extend Stein’s theory to a broader class of Gaussian random fields. We replace strict regular variation by more flexible O-regular variation, resolve the critical borderline case where the spectral exponent equals the dimension, and derive explicit convergence rates for prediction error. Our analysis also shows that screening is robust to mild nonstationarity and anisotropy, and it applies to irregular sampling designs such as Delone sets. To illustrate these results, we provide examples with Matérn, generalized Cauchy, and anisotropic models, along with numerical experiments confirming the theory. These findings clarify when local kriging can safely replace global prediction, and they provide a solid foundation for scalable methods such as covariance tapering and Vecchia approximations.