The Hausdorff dimension of multivariate operator-self-similar Gaussian random fields

TL;DR

This paper calculates the Hausdorff dimension of the range and graph of multivariate operator-self-similar Gaussian random fields, revealing it depends on eigenvalues of the scaling matrices.

Contribution

It solves an open problem by explicitly determining the Hausdorff dimension for these fields, linking it to eigenvalues and their multiplicities.

Findings

Hausdorff dimension depends on eigenvalues of E and D

Explicit formulas for the dimension of the range and graph

Eigenvalue multiplicities influence the dimension

Abstract

Let $\{X(t) : t \in \mathbb{R}^d \}$ be a multivariate operator-self-similar random field with values in $\mathbb{R}^m$. Such fields were introduced in [24] and satisfy the scaling property $\{X(c^E t) : t \in \mathbb{R}^d \} \stackrel{\rm d}{=} \{c^D X(t) : t \in \mathbb{R}^d \}$ for all $c > 0$, where $E$ is a $d \times d$ real matrix and $D$ is an $m \times m$ real matrix. We solve an open problem in [24] by calculating the Hausdorff dimension of the range and graph of a trajectory over the unit cube $K = [0,1]^d$ in the Gaussian case. In particular, we enlighten the property that the Hausdorff dimension is determined by the real parts of the eigenvalues of $E$ and $D$ as well as the multiplicity of the eigenvalues of $E$.