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

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 $E$.