A Bernstein-type inequality for \(U\)-statistics and \(U\)-processes (Q1347184)

A Bernstein-type inequality for the tail probability of non-degenerate \(U\)-statistics is given to complement the version for degenerate \(U\)- statistics given by the author and \textit{E. Giné} [Ann. Probab. 21, No. 3, 1494-1542 (1993; Zbl 0789.60031)]. This result improves on previous exponential inequalities in that, in the limit, the bound has the same order as the tail of the limiting distribution. Similar inequalities are established for the supremum of \(U\)-processes and these are applied to the local oscillations of the empirical distribution function of \(U\)- statistical structure.