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

\int f^{2} d P < \infty

Consider a set $ℱ \subset L_{P}^{2} (S)$ . 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) = \int f g d P - \int f d P \int g d P for f, g \in 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 $ℱ \subset L_{P}^{2} (S)$ is called pregaussian if for each $ω \in S,$ the function $f \mapsto G_{P} (f) (ω)$ on $ℱ$ is bounded, $ϱ_{P}$ -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

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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>
+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]]

Decimal128 floating-point format: Difference between revisions

Revision as of 20:06, 31 December 2013

Definition

Brownian bridge

References

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