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Concept
Pregaussian class
Summary
In probability theory, a pregaussian class or pregaussian set of functions is a set of functions, square integrable with respect to some probability measure, such that there exists a certain Gaussian process, indexed by this set, satisfying the conditions below.
For a probability space (S, Σ, P), denote by a set of square integrable with respect to P functions , that is
Consider a set . There exists a Gaussian process , indexed by , with mean 0 and covariance
Such a process exists because the given covariance is positive definite. This covariance defines a semi-inner product as well as a pseudometric on given by
Definition A class is called pregaussian if for each the function on is bounded, -uniformly continuous, and prelinear.
The process is a generalization of the brownian bridge. Consider with P being the uniform measure. In this case, the process indexed by the indicator functions , for is in fact the standard brownian bridge B(x). This set of the indicator functions is pregaussian, moreover, it is the Donsker class.
Official source
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