Operator Scaling Stable Random Fields

TL;DR

This paper introduces operator-scaling stable random fields, providing their moving average and harmonizable representations using E-homogeneous functions, and analyzes their sample path properties including Hölder exponents and Hausdorff dimension.

Contribution

It develops new representations for stable operator-scaling random fields and studies their sample path regularity and fractal properties.

Findings

Explicit moving average and harmonizable representations derived

Sample path Hölder exponents calculated for Gaussian case

Hausdorff dimension of sample paths determined

Abstract

A scalar valued random field is called operator-scaling if it satisfies a self-similarity property for some matrix E with positive real parts of the eigenvalues. We present a moving average and a harmonizable representation of stable operator scaling random fields by utilizing so called E-homogeneous functions. These fields also have stationary increments and are stochastically continuous. In the Gaussian case critical H\"{o}lder-exponents and the Hausdorff-dimension of the sample paths are also obtained.