Pregaussian class: Difference between revisions
No edit summary |
→References: add ISBN link |
||
Line 16: | Line 16: | ||
== References == |
== References == |
||
* R.M. Dudley |
*{{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:Probability theory]] |
Revision as of 21:07, 12 July 2009
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 , that is
Consider a set . There exists a Gaussian process , indexed by , with mean 0 and covariance
- for
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
Definition A class is called pregaussian if for each the function on is bounded, -uniformly continuous, and prelinear.
Brownian bridge
The process is a generalization of the brownian bridge. Consider with P being the uniform measure. In this case, the process indexed by the indicator functions , for 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