A Uniform Central Limit Theorem for Nonuniform $\phi$-Mixing Random Fields

Abstract

A sufficient condition is given for a sequence of partial-sum set-indexed processes with nonuniform $\phi$-mixing condition to converge to Brownian motion. The main result (Theorem 1.1) is an extension of the similar results of Goldie and Greenwood by weakening the $\phi$-mixing condition. An application (Corollary 4.2) to certain Gibbs fields is given.