Moderate deviations for degenerate U-processes

Sufficient conditions for a rank-dependent moderate deviations principle (MDP) for degenerate U-processes are presented. The MDP for VC classes of functions is obtained under exponential moments of the envelope. Among other techniques, randomization, decoupling inequalities and integrability of Gaussian and Rademacher chaos are used to present new Bernstein-type inequalities for U-processes which are the basis of our proofs of the MDP. We present a complete rank-dependent picture. The advantage of our approach is that we obtain in the degenerate case moderate deviations in non-Gaussian situations.