The Metric of Large Deviation Convergence

We construct a metric space of set functions ( Q(x), d) such that a sequence {P n} of Borel probability measures on a metric space ( X, d*) satisfies the full Large Deviation Principle (LDP) with speed {a n} and good rate function I if and only if the sequence {P an n} converges in ( Q(X), d) to the set function e –I . Weak convergence of probability measures is another special case of convergence in ( Q(X), d). Properties related to the LDP and to weak convergence are then characterized in terms of ( Q(x), d).