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Selfdecomposable Fields

By Ole E. Barndorff-Nielsen, Orimar Sauri and Benedykt Szozda
Thiele Research Reports
No. 02, February 2015

In the present paper we study selfdecomposability of random fields, as defined directly rather than in terms of finite-dimensional distributions. The main tools in our analysis are the master Lévy measure and the associated Lévy-Itô representation. We give the dilation criterion for selfdecomposability analogous to the classical one. Next, we give necessary and sufficient conditions (in terms of the kernel functions) for a Volterra field driven by a Lévy basis to be selfdecomposable. In this context we also study the so-called Urbanik classes of random fields. We follow this with the study of existence and selfdecomposability of integrated Volterra fields. Finally, we introduce infinitely divisible field-valued Lévy processes, give the Lévy-Itô representation associated with them and study stochastic integration with respect to such processes. We provide examples in the form of Lévy semistationary processes with a Gamma kernel and Ornstein-Uhlenbeck processes.

Keywords: selfdecomposability of random fields, Urbanik classes of random fields, random fields, Volterra fields

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