Moderate Deviations for Functional U-processes

The moderate deviations principle is shown for the partial sums processes built on U-empirical measures of Polish space valued random variables and on U-statistics of real valued kernel functions. It is proved that in the non-degenerate case the conditions for the time xed principles suuce for the moderate deviations principle to carry over to the corresponding partial sums processes. Given a uniformly bounded VC subgraph class of functions, we obtain corresponding moderate deviations for time dependent U-processes. Among other techniques we use the decoupling technique for U-statistics due to de la Pe~ na and Montgomery-Smith. We apply an improved version of a Bernstein-type inequality for degenerate U-statistics due to Arcones and Gin e as well as some moment inequalities for U-statistics. Moreover we prove and use a L evy-type maximal inequality for U-statistics.