## Bernoulli

• Bernoulli
• Volume 22, Number 1 (2016), 107-142.

### 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.

#### Article information

Source
Bernoulli, Volume 22, Number 1 (2016), 107-142.

Dates
Revised: February 2014
First available in Project Euclid: 30 September 2015

https://projecteuclid.org/euclid.bj/1443620845

Digital Object Identifier
doi:10.3150/14-BEJ624

Mathematical Reviews number (MathSciNet)
MR3449778

Zentralblatt MATH identifier
1339.60065

#### Citation

Kabluchko, Zakhar; Stoev, Stilian. Stochastic integral representations and classification of sum- and max-infinitely divisible processes. Bernoulli 22 (2016), no. 1, 107--142. doi:10.3150/14-BEJ624. https://projecteuclid.org/euclid.bj/1443620845

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