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 $\{X(c^{E}t):t\in {\mathbb{R}}^d\}\stackrel{\mathrm{d}}{=}\{c^{D}X(t):t\in {\mathbb{R}}^d\}$ 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 $E$.

References in corpus (3)

• Multivariate Operator-Self-Similar Random Fields
• Exact moduli of continuity for operator-scaling Gaussian random fields
• Modulus of continuity of some conditionally sub-Gaussian fields, application to stable random fields

Cited by in corpus (3)

• Fractal behavior of multivariate operator-self-similar stable random fields
• Wavelet Series Representation and Geometric Properties of Harmonizable Fractional Stable Sheets
• Sample path properties of generalized random sheets with operator scaling