On functional central limit theorem for stationary martingale random fields

Abstract

O. Introduction The term random field is often used to denote a collection of random variables with a parameter space which is a subset of the q-dimensional Euclidean space Rq. Stationary random fields are of great practical importance and hence also of theoretical interest. Examples of random fields occur in biological investigations concerning the distribution of plants or animals over a given area, when q=2 and t=(h, t2) is a point of the area. In problems involving propagation of electromaodaetic waves through random media the natural parameter space is a subset of R 4, representing space and time. Further important examples occur in the theory of turbulance where, for example, one may consider the case q=4 and t is a point in space-time, while r ~2(t), r are the velocity components of a turbulent fluid at the point t. Multiparameter stochastic process (the so-called random field) plays a prominent role in weak convergence of empirical process to Kiefer process (a two-dimensional Brownian bridge), Brownian sheets, and sample spacings. In this paper we extend the concept of martingale to random fields and obtain a functional central limit theorem for such random fields. An important example of martingales with a partially ordered parameter set is the following generalization of Wiener process. Let dq be the family of all Borel sets in Rq having finite Lebesgue measure. Let {X a, A ~d q} be a real Gaussian additive random set function with E(Xa) = 0, E(XaXB)=m (A A B) where m denotes the Lebesgue measure. Intuitively, Xa can be thought of as the integral over A of a Gaussian White noise. Such integral of Gaussian White noise has extensively been used by Physicists and engineers.