Rank-dependent moderate deviations of U-empirical measures in strong topologies

We prove a rank-dependent moderate deviation principle for U-empirical measures, where the underlying i.i.d. random variables take values in a measurable (not necessarily Polish) space (S,S). The result can be formulated on a suitable subset of all signed measures on (Sm,¢_m). We endow this space with a topology, which is stronger than the usual F-topology. A moderate deviation principle for Banach-space valued U-statistics is obtained as a particular application. The advantage of our result is that we obtain in the degenerate case moderate deviations in non-Gaussian situations with non-convex rate functions.