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

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

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

$\operatorname {Cov} (G_{P}(f),G_{P}(g))=EG_{P}(f)G_{P}(g)=\int fg\,dP-\int f\,dP\int g\,dP{\text{ for }}f,g\in {\mathcal {F}}$

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

$\varrho _{P}(f,g)=(E(G_{P}(f)-G_{P}(g))^{2})^{1/2}$

Definition A class ${\mathcal {F}}\subset L_{P}^{2}(S)$ 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.

References

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