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$$U$$-processes indexed by Vapnik-Červonenkis classes of functions with applications to asymptotics and bootstrap of $$U$$-statistics with estimated parameters. (English) Zbl 0807.62014
New results on exponential inequalities, on the law of the iterated logarithm and on the bootstrap central limit theorem for $$U$$-processes indexed by so-called $$VC$$ classes of functions [cf. R. M. Dudley, Lect. Notes Math. 1097, 1-142 (1984; Zbl 0554.60029)] are derived. These results are then applied to the asymptotics and the bootstrap of $$U$$- statistics with estimated parameters, in particular to the trimming of U- statistics. Earlier results are improved and extended.