Characterisations of set-indexed Brownian motion and associated conditions for finite-dimensional convergence
Let \({\mathcal A}\) be a subset of the \(\sigma\)-algebra of Borel subsets of \([0,1]^ d\). A process X on \({\mathcal A}\) is a collection \(\{X(A)\}_{A\in {\mathcal A}}\) of real-valued random variables. A standard Wiener process, or Brownian motion, on \({\mathcal A}\) is a Gaussian process W with the properties \(EW(A)=0\), \(cov(W(A),W(B))=| A\cap B|\) for A,B\(\in {\mathcal A}\), where \(| \cdot |\) is Lebesgue measure. The main results of the authors are characterizations of W on \({\mathcal R} = the\) class of all finite unions of left-open right-closed intervals in \([0,1]^ d\), and weak convergence to it. From these characterization and convergence results relative to an arbitrary class \({\mathcal A}\) are deduced. The results obtained are used to give new multidimensional central limit theorems for mixing random fields. They need only second- moment assumptions on the summands, a logarithmic mixing rate, and convergence of second moments of certain special partial sums.
scientific article; zbMATH DE number 3973946