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.
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 → R , that is
∫ f 2 d P < ∞ Consider a set F ⊂ L P 2 ( S ) . There exists a Gaussian process G P , indexed by F , with mean 0 and covariance
Cov ( G P ( f ) , G P ( g ) ) = E G P ( f ) G P ( g ) = ∫ f g d P − ∫ f d P ∫ g d P for f , g ∈ 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
ϱ P ( f , g ) = ( E ( G P ( f ) − G P ( g ) ) 2 ) 1 / 2 Definition A class F ⊂ L P 2 ( S ) is called pregaussian if for each ω ∈ S , the function f ↦ G P ( f ) ( ω ) on F is bounded, ϱ 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 ∈ [ 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.
