Large deviations of U-empirical measures in strong topologies and applications

We prove large deviation principles (LDP) for m-fold products of empirical measures and for U-empirical measures, where the underlying i.i.d: random variables take values in a measurable (not necessarily Polish) space (S, S). The results can be formulated on suitable subsets of all-probability measures on (S-m, S-xm). We endow the spaces with topologies, which are stronger than the usual tau-topology and which make integration with respect to certain unbounded; Banach-space valued functions a continuous operation: A special feature is the non-convexity of the rate function for m greater than or equal to 2. Improved versions of LDPs for Banach-space valued U- and V-statistics are obtained as a particular application. Some further applications concerning the Gibbs conditioning principle and a process level LDP are mentioned. (C) 2002 Editions scientifiques et medicales Elsevier SAS.