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 $L_P^2(S)$ a set of square integrable with respect to P functions $f:S\to R$, that is

$\int f^{2}dP<\mathrm{\infty }$

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

$\mathrm{Cov}(G_P(f),G_P(g))=E{G}_P(f)G_P(g)=\int fgdP-\int fdP\int gdP\text{ for }f,g\in {ℱ}$

Such a process exists because the given covariance is positive definite. This covariance defines a semi-inner product as well as a pseudometric on $L_P^2(S)$ given by

${\varrho }_P(f,g)=(E(G_P(f)-G_P(g){)}^2{)}^{1/2}$

Definition A class ${ℱ}\subset L_P^2(S)$ is called pregaussian if for each $\omega \in S,$ the function $f\mapsto G_P(f)(\omega )$ on ${ℱ}$ 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.

References

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