The Annals of Probability

Ergodic properties of Poissonian ID processes

Emmanuel Roy

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Abstract

We show that a stationary IDp process (i.e., an infinitely divisible stationary process without Gaussian part) can be written as the independent sum of four stationary IDp processes, each of them belonging to a different class characterized by its Lévy measure. The ergodic properties of each class are, respectively, nonergodicity, weak mixing, mixing of all order and Bernoullicity. To obtain these results, we use the representation of an IDp process as an integral with respect to a Poisson measure, which, more generally, has led us to study basic ergodic properties of these objects.

Article information

Source
Ann. Probab., Volume 35, Number 2 (2007), 551-576.

Dates
First available in Project Euclid: 30 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1175287754

Digital Object Identifier
doi:10.1214/009117906000000692

Mathematical Reviews number (MathSciNet)
MR2308588

Zentralblatt MATH identifier
1146.60031

Subjects
Primary: 60G10: Stationary processes 60E07: Infinitely divisible distributions; stable distributions 37A05: Measure-preserving transformations
Secondary: 37A40: Nonsingular (and infinite-measure preserving) transformations 60G55: Point processes

Keywords
Infinitely divisible stationary processes Poisson suspensions ergodic theory infinite-measure preserving transformations

Citation

Roy, Emmanuel. Ergodic properties of Poissonian ID processes. Ann. Probab. 35 (2007), no. 2, 551--576. doi:10.1214/009117906000000692. https://projecteuclid.org/euclid.aop/1175287754


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