Lévy's Brownian motion as a set-indexed process and a related central limit theorem

The Brownian motion with multi-dimensional time parameter introduced by Paul Lévy can be viewed as a set-indexed Brownian process with independent increments. This is demonstrated in a way which yields a unified representation of Lévy's Brownian motion and the Brownian sheet. Lévy's Brownian motion, like Brownian sheet, is shown to be a special case of the additive set-indexed Gaussian process {Z(A): A [epsilon] A} with Cov(Z(A), Z(B) =[mu](A [intersection] B) for some measure [mu]. A particular family of spheres is seen to play the same basic role in this representation as the family of orthants plays for Brownian sheet. A related central limit theorem and invariance result are discussed for a natural family of empirical-like processes, indexed by large families of sets A.