Representations, decompositions and sample function continuity of random fields with independent increments

Abstract

Standard fare in the study of representations and decompositions of processes with independent increments is pursued in the somewhat more complex setting of vector-valued random fields having independent increments over disjoint sets. Such processes are first constructed as almost surely uniformly convergent sums of Poisson type summands, that immediately yield information on sample function properties of versions. The constructions employed, which include a generalized version of the Ferguson-Klass construction with uniform convergence, are new even in the simpler setting of processes in one-dimensional time. Following these constructions, or representations, an analogue of the Lévy-Ito decomposition for Lévy processes is developed, which then enables a number of simple sample function properties of these processes to be read off from the Lévy measure in their characteristic functionals. The paper concludes with a study of general centred additive random fields and an appendix incorporating a brief survey of the theory of centred sums of independent random variables.