Rank-dependent Moderate Deviations of U-empirical Measures in Strong Topologies∗

We prove a rank-dependent moderate deviation principle for Uempirical 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,S⊗m). We endow this space with a topology, which is stronger than the usual τ -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.