Moderate and large deviations for U-processes

Sufficient conditions for the moderate and large deviation principle for U-processes are given. For the large deviation result the conditions are in terms of “blockwise” empirical process conditions. On the moderate scaling the case of U-processes indexed by a uniformly bounded VC subgraph class of functions is considered. The proofs are based on an isoperimetric inequality for empirical processes due to Talagrand, a truncation method based on an isoperimetric inequality by Ledoux, the existence of almost regular partitions of complete hypergraphs due to Baranyai and a Bernstein-type inequality for U-processes due to Arcones and Giné.