Stone-Weierstrass type theorems for large deviations
Henri Comman, University of Santiago de Chile
Abstract
We give a general version of Bryc's theorem valid on any topological
space and with any algebra $mathcal{A}$ of real-valued continuous
functions separating the points, or any well-separating class. In
absence of exponential tightness, and when the underlying space is
locally compact regular and $mathcal{A}$ constituted by functions
vanishing at infinity, we give a sufficient condition on the
functional $Lambda(cdot)_{mid mathcal{A}}$ to get large
deviations with not necessarily tight rate function. We obtain the
general variational form of any rate function on a completely
regular space; when either exponential tightness holds or the space
is locally compact Hausdorff, we get it in terms of any algebra as
above. Prohorov-type theorems are generalized to any space, and
when it is locally compact regular the exponential tightness
can be replaced by a (strictly weaker) condition on
$Lambda(cdot)_{mid mathcal{A}}$.
Full text: PDF | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona.
The preferred way is to send a scanned (jpeg or pdf) copy of the signed copyright form to the managing editor Philippe Carmona at ejpecpme@math.univ-nantes.fr.
If a paper has several authors, the corresponding author signs the copyright form
on behalf of all the authors.