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

Mina Ossiander and Ronald Pyke

Stochastic Processes and their Applications, 1985, vol. 21, issue 1, 133-145

Abstract: 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.

Keywords: weak; convergence; Gaussian; processes; white-noise; invariance; principle; Lévy's; Brownian; motion; empirical; processes (search for similar items in EconPapers)
Date: 1985
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Citations: View citations in EconPapers (3)

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