Functional Approach of Large Deviations in General Spaces

Henri Comman ()
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Henri Comman: University of Santiago de Chile

Journal of Theoretical Probability, 2005, vol. 18, issue 1, 187-207

Abstract: Let X be a topological space, (μ α) a net of Borel probability measures on X, and (tα) a net in ]0,∞[ converging to 0. Let $$\cal A$$ be a set of continuous functions such that for all x ∈X that can be suitably distinguished by some continuous functions from any closed set not containing $$x, \cal A$$ contains such a distinguishing function. Assuming that $$\Lambda(h) = \log \lim\left(\int_{X} e^{h(x)/t_{\alpha}} \mu _{\alpha}(dx)\right)^{t_{\alpha}}$$ exists for all $$h \in \cal A$$ , we give a sufficient condition in order that (μ α) satisfies a large deviation principle with powers (tα) and not necessary tight rate function. When X is completely regular (not necessary Hausdorff), this condition is also necessary, and so strictly weaker than exponential tightness; this allows us to strengthen Bryc’s theorem in various ways. We give the general form of a rate function in terms of $$\cal A$$ . A Prohorov-type theorem with a weaker notion than exponential tightness is obtained, which improves known results.

Keywords: Large deviations; converse Varadhan’s theorem problem (search for similar items in EconPapers)
Date: 2005
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DOI: 10.1007/s10959-004-2594-2

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