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 (S, &Sigma;, P), denote by $$L^2_P(S)$$ a set of square integrable with respect to P functions $$f:S\to R$$, that is


 * $$ \int f^2 \, dP<\infty$$

Consider a set $$\mathcal{F}\subset L^2_P(S)$$. There exists a Gaussian process $$G_P$$, indexed by $$\mathcal{F}$$, with mean 0 and covariance


 * $$\operatorname{Cov} (G_P(f),G_P(g))= E G_P(f)G_P(g)=\int fg\, dP-\int f\,dP \int g\,dP\text{ for }f,g\in\mathcal{F}$$

Such a process exists because the given covariance is positive definite. This covariance defines a semi-inner product as well as a pseudometric on $$L^2_P(S)$$ given by
 * $$\varrho_P(f,g)=(E(G_P(f)-G_P(g))^2)^{1/2}$$

Definition A class $$\mathcal{F}\subset L^2_P(S)$$ is called pregaussian if for each $$\omega\in S,$$ the function $$f\mapsto G_P(f)(\omega)$$ on $$\mathcal{F}$$ is bounded, $$\varrho_P$$-uniformly continuous, and prelinear.

Brownian bridge
The $$G_P$$ process is a generalization of the brownian bridge. Consider $$S=[0,1],$$ with P being the uniform measure. In this case, the $$G_P$$ process indexed by the indicator functions $$I_{[0,x]}$$, for $$x\in [0,1],$$ is in fact the standard brownian bridge B(x). This set of the indicator functions is pregaussian, moreover, it is the Donsker class.