Pregaussian class

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

For a probability space, 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

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. This set of the indicator functions is pregaussian, moreover, it is the Donsker class.