Random Fields and Geometry pp 65-74 | Cite as

# Orthogonal Expansions

Chapter

## Abstract

While most of what we shall have to say in this brief chapter is rather theoretical, it actually covers one of the most important practical aspects of Gaussian modeling. The basic result is Theorem 3.1.1, which states that

*every*centered Gaussian process with a continuous covariance function has an expansion of the form where the ξ_{n}are i.i.d.*N*(0, 1), and the ϕn are certain functions on*T*determined by the covariance function*C*of*f*. In general, the convergence in (3.0.1) is in*L*^{2}({ℙ) for each*t*∈*T*, but (Theorem 3.1.2) if*f*is a.s. continuous then the convergence is uniform over*T*, with probability one.## Keywords

Orthonormal Basis Covariance Function Reproduce Kernel Hilbert Space Standard Brownian Motion Orthogonal Expansion
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media LLC 2007