Pregaussian class explained

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

a set of square integrable with respect to P functions

f:S\toR

, that is

\intf²dP<infty

Consider a set

. There exists a Gaussian process

G_P

, indexed by

l{F}

, with mean 0 and covariance

\operatorname{Cov}(G_P(f),G_P(g))=EG_P(f)G_P(g)=\intfgdP-\intfdP\intgdPforf,g\inl{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

given by

\varrho_P(f,g)=(E(G_P(f)-G

^1/2

Definition A class

is called pregaussian if for each

\omega\inS,

the function

f\mapstoG_P(f)(\omega)

l{F}

is bounded,

\varrho_P

-uniformly continuous, and prelinear.

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