Operator-stable and operator-self-similar random fields

TL;DR

This paper introduces two classes of multivariate random fields with operator-stable marginals that are invariant under operator-scaling, constructed using operator-stable random measures and homogeneous functions.

Contribution

It constructs new classes of operator-stable and operator-self-similar random fields with specific invariance properties, expanding the theory of multivariate stable processes.

Findings

Defined operator-stable random fields with operator-scaling invariance

Constructed fields using operator-stable random measures and homogeneous functions

Provided mathematical framework for multivariate operator-stable processes

Abstract

Two classes of multivariate random fields with operator-stable marginals are constructed. The random fields $\mathbb{X}=\{X(t) : t \in \mathbb{R}^d \}$ with values in $\mathbb{R}^m$ are invariant in law under operator-scaling in both the time-domain and the state-space. The construction is based on operator-stable random measures utilising certain homogeneous functions.