Selfdecomposable Fields

TL;DR

This paper investigates the selfdecomposability of random fields using the master Lévy measure and Lévy-Itô representation, providing criteria, conditions, and examples for various classes of infinitely divisible fields.

Contribution

It introduces new criteria and conditions for selfdecomposability of random fields, including Volterra and Lévy-driven fields, and develops the theory of infinitely divisible field-valued Lévy processes.

Findings

Dilation criterion for selfdecomposability of fields.

Necessary and sufficient conditions for Volterra fields.

Examples of Lévy semistationary and Ornstein-Uhlenbeck processes.

Abstract

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 L\'evy measure and the associated L\'evy-It\^o 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 L\'evy 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 L\'evy processes, give the L\'evy-It\^o representation associated with them and study stochastic integration with respect to such processes. We provide examples in the form of L\'evy semistationary processes with a Gamma kernel and Ornstein-Uhlenbeck processes.