Abstract
In this work we present different results concerning mixing properties of multivariate infinitely divisible (ID) stationary random fields. First, we derive some necessary and sufficient conditions for mixing of stationary ID multivariate random fields in terms of their spectral representation. Second, we prove that (linear combinations of independent) mixed moving average fields are mixing. Further, using a simple modification of the proofs of our results, we are able to obtain weak mixing versions of our results. Finally, we prove the equivalence of ergodicity and weak mixing for multivariate ID stationary random fields.
Highlights
In 1970 in his fundamental work [11], Maruyama provided pivotal results for infinitely divisible (ID) processes
In 1996 Rosinski and Zak extended Maruyama results proving that the a stationary ID process (Xt )t∈R is mixing if and only if limt→∞ E ei(Xt −X0) = E ei X0 E e−i X0, provided the Lévy measure of X0 has no atoms in 2π Z
On the modelling/application level, we prove that multivariate mixed moving average fields are mixing
Summary
In 1970 in his fundamental work [11], Maruyama provided pivotal results for infinitely divisible (ID) processes. Journal of Theoretical Probability (2019) 32:1845–1879 results of Rosinski and Zak to the multivariate case Parallel to this line of research, new developments have been obtained for ergodic and weak mixing properties of infinitely divisible random fields. In the present work we fill an important gap by extending the results of Maruyama [11], Rosinski and Zak [13,14], and Fuchs and Stelzer [6] to the multivariate random field case. This is crucial for applications since many of them consider a multidimensional domain composed by both spatial and temporal components (and not just temporal ones). In order to simplify the exposition, we decided to put long proofs in the appendices
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