Fractal behavior of multivariate operator-self-similar stable random fields

TL;DR

This paper studies the sample path regularity of multivariate operator-self-similar stable random fields, extending previous work to cases where the scaling matrices differ from identity, thus solving an open problem.

Contribution

It provides the first analysis of sample path properties for these fields with non-identity scaling matrices, advancing understanding of their regularity.

Findings

Sample path regularity characterized for multivariate operator-self-similar stable fields.

First results for fields with non-identity scaling matrices.

Addresses an open problem in the literature.

Abstract

We investigate the sample path regularity of multivariate operator-self-similar stable random fields with values in $\mathbb{R}^m$ given by a harmonizable representation. Such fields were introduced in [25] as a generalization of both operator-self-similar stochastic processes and operator scaling random fields and satisfy the scaling property $\{X(c^E t) : t \in \mathbb{R}^d \} \stackrel{\rm d}{=} \{c^D X(t) : t \in \mathbb{R}^d \}$, where $E$ is a real $d \times d$ matrix and $D$ is a real $m \times m$ matrix. This paper provides the first results concerning sample path properties of such fields, including both $E$ and $D$ different from identity matrices. In particular, this solves an open problem in [25].