Characterisations of Set-Indexed Brownian Motion and Associated Conditions for Finite-Dimensional Convergence

Abstract

Two characterisations are given of the finite-dimensional laws of Brownian motion indexed by an arbitrary class of subsets of the $d$-dimensional unit cube. There are associated conditions for convergence of finite-dimensional laws of a sequence of set-indexed additive processes. These conditions have a more explicit form in the case of set-indexed partial-sum processes based on mixing random variables.