Stochastic integral representations and classification of sum- and max-infinitely divisible processes

Abstract

Introduced is the notion of minimality for spectral representations of sum- and max-infinitely divisible processes and it is shown that the minimal spectral representation on a Borel space exists and is unique. This fact is used to show that a stationary, stochastically continuous, sum- or max-i.d. random process on $\mathbb{R}^{d}$ can be generated by a measure-preserving flow on a $\sigma$-finite Borel measure space and that this flow is unique. This development makes it possible to extend the classification program of Rosiński (Ann. Probab. 23 (1995) 1163–1187) with a unified treatment of both sum- and max-infinitely divisible processes. As a particular case, a characterization of stationary, stochastically continuous, union-infinitely divisible random measurable subsets of $\mathbb{R}^{d}$ is obtained. Introduced and classified are several new max-i.d. random field models including fields of Penrose type and fields associated to Poisson line processes.