Uniform dimension results for Gaussian random fields

Abstract

Let X = {X(t), t ∈ ℝN} be a Gaussian random field with values in ℝd defined by $$ X(t) = (X_1 (t),...,X_d (t)),\forall t \in \mathbb{R}^N . $$ (1) . The properties of space and time anisotropy of X and their connections to uniform Hausdorff dimension results are discussed. It is shown that in general the uniform Hausdorff dimension result does not hold for the image sets of a space-anisotropic Gaussian random field X.