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December, 1980 Isotropic Gaussian Processes on the Hilbert Sphere
Simeon M. Berman
Ann. Probab. 8(6): 1093-1106 (December, 1980). DOI: 10.1214/aop/1176994571


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 $n \rightarrow \infty$.


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Simeon M. Berman. "Isotropic Gaussian Processes on the Hilbert Sphere." Ann. Probab. 8 (6) 1093 - 1106, December, 1980.


Published: December, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0452.60045
MathSciNet: MR602383
Digital Object Identifier: 10.1214/aop/1176994571

Primary: 60G15
Secondary: 60F05 , 60F15

Keywords: central limit theorem , determinism , ergodic theorem , harmonic property , Hilbert sphere , Isotropic Gaussian process

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 6 • December, 1980
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