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Pregaussian class

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

∫ f 2 d P < ∞ {\displaystyle \int f^{2}\,dP<\infty }

Consider a set F ⊂ L P 2 ( S ) {\displaystyle {\mathcal {F}}\subset L_{P}^{2}(S)} . There exists a Gaussian process G P {\displaystyle G_{P}} , indexed by F {\displaystyle {\mathcal {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 {\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 L P 2 ( S ) {\displaystyle L_{P}^{2}(S)} given by

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

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

Brownian bridge

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