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, Σ, P), denote by [math]\displaystyle{ L^2_P(S) }[/math] a set of square integrable with respect to P functions [math]\displaystyle{ f:S\to R }[/math], that is

[math]\displaystyle{ \int f^2 \, dP\lt \infty }[/math]

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

[math]\displaystyle{ \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} }[/math]

Such a process exists because the given covariance is positive definite. This covariance defines a semi-inner product as well as a pseudometric on [math]\displaystyle{ L^2_P(S) }[/math] given by

[math]\displaystyle{ \varrho_P(f,g)=(E(G_P(f)-G_P(g))^2)^{1/2} }[/math]

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

Brownian bridge

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

References

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