Pregaussian class

Definition

For a probability space (S, Σ, 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

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.

References