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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Criteria for large deviations


Author: Henri Comman
Journal: Trans. Amer. Math. Soc. 355 (2003), 2905-2923
MSC (2000): Primary 60F10
Published electronically: March 17, 2003
MathSciNet review: 1975405
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Abstract: We give the general variational form of

\begin{displaymath}\limsup(\int_X e^{h(x)/t_{\alpha}}\mu_{\alpha}(dx))^{t_{\alpha}}\end{displaymath}

for any bounded above Borel measurable function $h$ on a topological space $X$, where $(\mu_{\alpha})$ is a net of Borel probability measures on $X$, and $(t_{\alpha})$ a net in $]0,\infty[$ converging to $0$. When $X$ is normal, we obtain a criterion in order to have a limit in the above expression for all $h$ continuous bounded, and deduce new criteria of a large deviation principle with not necessarily tight rate function; this allows us to remove the tightness hypothesis in various classical theorems.


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Additional Information

Henri Comman
Affiliation: Department of Mathematics, University of Santiago of Chile, Bernardo O’Higgins 3363, Santiago, Chile
Email: hcomman@usach.cl

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03274-4
PII: S 0002-9947(03)03274-4
Received by editor(s): January 3, 2002
Received by editor(s) in revised form: November 9, 2002
Published electronically: March 17, 2003
Additional Notes: This work was supported in part by FONDECYT Grant 3010005
Article copyright: © Copyright 2003 American Mathematical Society