On the orthogonality of zero-mean Gaussian measures: Sufficiently dense sampling

TL;DR

This paper investigates conditions under which two zero-mean Gaussian measures, derived from stationary random functions, are either equivalent or orthogonal, focusing on the role of dense sampling in the domain.

Contribution

It provides an isotropic analog for Gaussian measure equivalence and establishes that orthogonality can be inferred from dense sampling of distances in the domain.

Findings

Equivalence linked to square-integrable extension of covariance differences

Orthogonality can be deduced from dense set of distances

Results extend classical Gaussian measure theory to isotropic cases

Abstract

For a stationary random function $\xi$, sampled on a subset $D$ of $\mathbb{R}^{d}$, we examine the equivalence and orthogonality of two zero-mean Gaussian measures $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ associated with $\xi$. We give the isotropic analog to the result that the equivalence of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ is linked with the existence of a square-integrable extension of the difference between the covariance functions of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ from $D$ to $\mathbb{R}^{d}$. We show that the orthogonality of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ can be recovered when the set of distances from points of $D$ to the origin is dense in the set of non-negative real numbers.