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.

Definition

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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.

Brownian bridge

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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.

References

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  • R. M. Dudley (1999), Uniform central limit theorems, Cambridge, UK: Cambridge University Press, p. 436, ISBN 0-521-46102-2