Isotropic Gaussian Processes on the Hilbert Sphere

Abstract

The subject of this work is a study of four properties of an isotropic Gaussian process on an infinite dimensional sphere in Hilbert space. The process is deterministic in the sense that its values on an arbitrary nonempty open subset of the sphere determine its values throughout the sphere. An harmonic property is defined, and is characterized in terms of the covariance function of the process. If $f$ is a function of a real variable which is square-integrable with respect to the Gaussian density, then the average of $f$ over the values of the process on an $n$-dimensional subsphere converges with probability 1 for $n \rightarrow \infty$. Under further conditions on the process the average of $f$ has, with appropriate normalization, a limiting Gaussian distribution for $n \rightarrow \infty$.