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.
Categories: math.PR