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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'', &Sigma;, ''P''), denote by <math>L^2_P(S)</math> a set of square integrable with respect to ''P'' functions <math>f:S\to R</math>, that is


:<math> \int f^2 \, dP<\infty</math>
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Consider a set <math>\mathcal{F}\subset L^2_P(S)</math>. There exists a [[Gaussian process]] <math>G_P</math>, indexed by <math>\mathcal{F}</math>, with mean 0 and covariance
 
:<math>\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}</math>
Such a process exists because the given covariance is positive definite. This covariance defines a semi-inner product as well as a [[pseudometric space|pseudometric]] on <math>L^2_P(S)</math> given by
:<math>\varrho_P(f,g)=(E(G_P(f)-G_P(g))^2)^{1/2}</math>
 
'''Definition''' A class <math>\mathcal{F}\subset L^2_P(S)</math> is called '''pregaussian''' if for each <math>\omega\in S,</math> the function <math>f\mapsto G_P(f)(\omega)</math> on <math>\mathcal{F}</math> is bounded, <math>\varrho_P</math>-uniformly continuous, and prelinear.
 
==Brownian bridge==
The <math>G_P</math> process is a generalization of the [[brownian bridge]]. Consider <math>S=[0,1],</math> with ''P'' being the [[uniform distribution (continuous)|uniform measure]]. In this case, the <math>G_P</math> process indexed by the [[indicator function]]s <math>I_{[0,x]}</math>, for <math>x\in [0,1],</math> is in fact the standard [[brownian bridge]] ''B''(''x''). This set of the indicator functions is pregaussian, moreover, it is the [[Donsker class]].
 
== References ==
*{{citation | author = R. M. Dudley | title = Uniform central limit theorems | publisher = Cambridge University Press | location = Cambridge, UK | year = 1999 | pages = 436 | isbn = 0-521-46102-2}}
 
[[Category:Probability theory]]
[[Category:Empirical process]]

Latest revision as of 18:47, 11 August 2014


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