Ergodic properties of Poissonian ID processes

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.

Published at http://dx.doi.org/10.1214/009117906000000692 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (16)

• Ergodic properties of sum- and max-stable stationary random fields via null and positive group actions
• Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows
• Stochastic integral representations and classification of sum- and max-infinitely divisible processes
• Maxima of long memory stationary symmetric $\alpha$-stable processes, and self-similar processes with stationary max-increments
• Maharam extension and stationary stable processes
• Spectral multiplicities for infinite measure preserving transformations
• Symmetric Stable Processes on Amenable Groups
• Mixing constructions with infinite invariant measure and spectral multiplicities
• Mixing of Poisson random measures under interacting transformations
• A functional large and moderate deviation principle for infinitely divisible processes driven by null-recurrent markov chains
• On the Zero-Type property and mixing of Bernoulli shifts
• Ergodic Properties of Max-Infinitely Divisible Processes
• Limit Theory for the Sample Autocovariance for Heavy Tailed Stationary Infinitely Divisible Processes Generated by Conservative Flows
• Nonsingular Group Actions and Stationary S alpha S Random Fields
• Group measure space construction, ergodicity and $W^{\ast }$-rigidity for stable random fields
• Decomposability for stable processes