Large Deviations of Products of Empirical Measures and U-empirical Measures in Strong Topologies

We prove a large deviation principle (LDP) for m-fold products of empirical measures and for U -empirical measures, where the state space (S,S) of the underlying i. i. d. sequence of random variables is an arbitrary measurable space. For this LDP we choose a suitable subset MΦ(S ) of all probability measures on (S,S) and endow it with a topology, which is stronger than the τ -topology and makes the map ν 7→ ∫ Sm f dν continuous even for certain unbounded f . An improved form of a LDP for U and V statistics is obtained as a particular application. Furthermore, we improve the Gibbs conditioning principle for interacting ensembles of particles.