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 Electronic Communications in Probability > Vol. 13 (2008) > Paper 22 open journal systems 

Stone-Weierstrass type theorems for large deviations

Henri Comman, University of Santiago de Chile

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 $Lambda(cdot)_{mid mathcal{A}}$.

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Pages: 225-240

Published on: April 28, 2008

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Electronic Communications in Probability. ISSN: 1083-589X