Functional Approach of Large Deviations in General Spaces

Let X be a topological space, ( ) a net of Borel probability measures on X, and (t) a net in ]0, converging to 0. Let 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 contains such a distinguishing function. Assuming that exists for all , we give a sufficient condition in order that ( ₎ satisfies a large deviation principle with powers (t₎ 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 Brycs theorem in various ways. We give the general form of a rate function in terms of . A Prohorov-type theorem with a weaker notion than exponential tightness is obtained, which improves known results.