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2008 Stone-Weierstrass type theorems for large deviations
Henri Comman
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Electron. Commun. Probab. 13: 225-240 (2008). DOI: 10.1214/ECP.v13-1370


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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Henri Comman. "Stone-Weierstrass type theorems for large deviations." Electron. Commun. Probab. 13 225 - 240, 2008.


Accepted: 28 April 2008; Published: 2008
First available in Project Euclid: 6 June 2016

zbMATH: 1189.60056
MathSciNet: MR2399284
Digital Object Identifier: 10.1214/ECP.v13-1370

Primary: 60F10


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