Functional Approach of Large Deviations in General Spaces

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal A$$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x, \cal A$$\end{document} contains such a distinguishing function. Assuming that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda(h) = \log \lim\left(\int_{X} e^{h(x)/t_{\alpha}} \mu _{\alpha}(dx)\right)^{t_{\alpha}}$$\end{document} exists for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h \in \cal A$$\end{document}, 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal A$$\end{document}. A Prohorov-type theorem with a weaker notion than exponential tightness is obtained, which improves known results.