Practically Applicable Central Limit Theorem for Spatial Statistics

Abstract

Let {Z(s):s∈D⊆ℝ d } be a zero mean stationary random field observed at a finite number of locations. Lahiri (Sankhya Ser. A 65:356–388, 2003) proved spatial central limit theorems (CLT) for ∑ =1 Z(s i ) assuming a ‘nearly infill domain sampling’. Applications of his results depended on the underlying spatial sampling region and the design in a complicated fashion. The main objective of this paper is to provide CLTs that could be applied easily in practice. We present two main results assuming a ‘nearly infill domain sampling’ defined mainly in terms of dependence. Theorem 1 establishes a CLT for ∑ =1 Z(s i ) and Theorem 2 is obtained mainly for applications to density estimates. We report on a simulation study for illustrating a way of applying our results in practice.