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Ergodic properties of Poissonian ID processes. (English) Zbl 1146.60031
A general infinitely divisible (ID) process is an independent sum of a Gaussian process and a Poissonian (IDp) process, the latter being uniquely characterized by its Lévy measure. In the paper, it is shown that every stationary IDp process can be uniquely decomposed in the independent sum of four IDp processes which are non-ergodic, weakly mixing, mixing of all order and Bernoulli, respectively. Furthermore, an explicit form of a stationary IDp process with a dissipative Lévy measure is given. If this process is square integrable, some spectral criteria for its ergodic behaviour are established. If the IDp process is \(\alpha\)-stable, the four components of its decomposition are \(\alpha\)-stable as well. The proofs are mainly based on the ideas from the ergodic theory of dynamical systems, e.g. on proper decompositions of an invariant measure.

MSC:
60G10 Stationary stochastic processes
60E07 Infinitely divisible distributions; stable distributions
37A05 Dynamical aspects of measure-preserving transformations
37A40 Nonsingular (and infinite-measure preserving) transformations
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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