Central limit theorems and uniform laws of large numbers for arrays of random fields

Abstract

Spatial-interaction models are being increasingly considered in economics, and have a long tradition in geography, regional science and urban economics. In this paper, we derive a new central limit theorem, a new law of large numbers and a new uniform law of large numbers for spatial processes, or random fields. Such limit theorems form the essential building blocks towards developing an asymptotic theory of M-estimators for spatial processes, including maximum likelihood and generalized method of moments estimators. The development of a general estimation theory has been hampered by lack of general limit theo-rems. In this paper, we establish limit theorems that are applicable to a broad range of data processes in economics and other fields. In particular, we extend the literature by considering weakly dependent random fields located on arbi-trary unevenly spaced lattices in d-dimensional Euclidean space, and allow for spatial processes that are non-stationary, possibly with unbounded moments. We provide weak, yet primitive, sufficient conditions for each of the theorems