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Stone-Weierstrass type theorems for large deviations. (English) Zbl 1189.60056

Summary: 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 \(\Lambda(\cdot)_{|{\mathcal A}}\) 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 \(\Lambda(\cdot)_{|{\mathcal A}}\).

MSC:

60F10 Large deviations
46J10 Banach algebras of continuous functions, function algebras
46N30 Applications of functional analysis in probability theory and statistics
60B05 Probability measures on topological spaces
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