Moderate deviations for functional U-processes

The moderate deviations principle is shown for the partial sums processes built of U-empirical measures and of U-statistics. It is proved that in the non-degenerate case the conditions for the fixed time principles suffice 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. We use decoupling techniques and apply an improved version of a Bernstein-type inequality for degenerate U-statistics. Moreover, we prove and use a Levy-type maximal inequality for U-statistics, (C) 2001 Editions scientifiques et medicales Elsevier SAS.