Norm-dependent covariance permissibility of weakly homogeneous spatial random fields and its consequences in spatial statistics

Permissibility of a covariance function (in the sense of Bochner) depends o n the norm (or metric) that determines spatial distance in several dimensio ns. A covariance function that is permissible for one norm may not be so fo r another. We prove that for a certain class of covariances of weakly homog eneous random fields, the spatial distance can be defined only in terms of the Euclidean norm. This class includes commonly used covariance functions. Functions that do not belong to this class may be permissible covariances for some non-Euclidean metric. Thus, a different class of covariances, for which non-Euclidean norms are valid spatial distances, is also discussed. T he choice of a coordinate system and associated norm to describe a physical phenomenon depends on the nature of the properties being described. Norm-d ependent permissibility analysis has important consequences in spatial stat istics applications (e.g., spatial estimation or mapping), in which one is concerned about the validity of covariance functions associated with a phys ically meaningful norm (Euclidean or non-Euclidean).