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

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

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

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

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

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

Definition A class ${\displaystyle {\mathcal {F}}\subset L_{P}^{2}(S)}$ is called pregaussian if for each ${\displaystyle \omega \in S,}$ the function ${\displaystyle f\mapsto G_{P}(f)(\omega )}$ on ${\displaystyle {\mathcal {F}}}$ is bounded, ${\displaystyle \varrho _{P}}$-uniformly continuous, and prelinear.

Brownian bridge

The ${\displaystyle G_{P}}$ process is a generalization of the brownian bridge. Consider ${\displaystyle S=[0,1],}$ with P being the uniform measure. In this case, the ${\displaystyle G_{P}}$ process indexed by the indicator functions ${\displaystyle I_{[0,x]}}$, for ${\displaystyle x\in [0,1],}$ 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