A Bernstein-type inequality for U-statistics and U-processes

A Bernstein-type inequality for non-degenerated U-statistics is presented. As the Bernstein inequality for sums of independent identically distributed random variables, in the limit, its tail has the same order as the tail of the limit. We also consider the case of U-processes indexed by a uniformly bounded VC subgraph class of functions. This is applied to obtain exponential inequalities for the local oscillations of the empirical distribution function of U-statistical structure.