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February 2016 Stochastic integral representations and classification of sum- and max-infinitely divisible processes
Zakhar Kabluchko, Stilian Stoev
Bernoulli 22(1): 107-142 (February 2016). DOI: 10.3150/14-BEJ624


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.


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Zakhar Kabluchko. Stilian Stoev. "Stochastic integral representations and classification of sum- and max-infinitely divisible processes." Bernoulli 22 (1) 107 - 142, February 2016.


Received: 1 July 2012; Revised: 1 February 2014; Published: February 2016
First available in Project Euclid: 30 September 2015

zbMATH: 1339.60065
MathSciNet: MR3449778
Digital Object Identifier: 10.3150/14-BEJ624

Keywords: infinitely divisible process , max-infinitely divisible process , measure-preserving flow , minimality , Poisson process , ‎spectral representation , stochastic integral

Rights: Copyright © 2016 Bernoulli Society for Mathematical Statistics and Probability

Vol.22 • No. 1 • February 2016
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