Spectral conditions for strong local nondeterminism and exact Hausdorff measure of ranges of Gaussian random fields

TL;DR

This paper establishes spectral conditions under which Gaussian random fields exhibit strong local nondeterminism and determines their exact Hausdorff measure of the range, extending previous theorems and broadening applicability.

Contribution

It provides a new spectral condition for strong local nondeterminism and calculates the exact Hausdorff measure for the range of Gaussian fields with stationary increments.

Findings

Spectral conditions for strong local nondeterminism derived.

Exact Hausdorff measure function for the range determined.

Extended applicability beyond previous theorems.

Abstract

Let $X= \{X(t), t \in \R^N\}$ be a Gaussian random field with values in $\R^d$ defined by \[ X(t) = \big(X_1(t),..., X_d(t)\big),\qquad t \in \R^N, \] where $X_1, ..., X_d$ are independent copies of a real-valued, centered, anisotropic Gaussian random field $X_0$ which has stationary increments and the property of strong local nondeterminism. In this paper we determine the exact Hausdorff measure function for the range $X([0, 1]^N)$. We also provide a sufficient condition for a Gaussian random field with stationary increments to be strongly locally nondeterministic. This condition is given in terms of the spectral measures of the Gaussian random fields which may contain either an absolutely continuous or discrete part. This result strengthens and extends significantly the related theorems of Berman (1973, 1988), Pitt (1978) and Xiao (2007, 2009), and will have wider applicability beyond the scope of the present paper.