On the Central Limit Theorem for Nonuniform φ-Mixing Random Fields

Abstract

The partial-sum processes, indexed by sets, of a stationary nonuniform φ-mixing random field on the d-dimensional integer lattice are considered. A moment inequality is given from which the convergence of the finite-dimensional distributions to a Brownian motion on the Borel subsets of [0, 1]d is obtained. A Uniform CLT is proved for classes of sets with a metric entropy restriction and applied to certain Gibbs fields. This extends some results of Chen(5) for rectangles. In this case and when the variables are bounded a simpler proof of the uniform CLT is given.