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A notion of local $U$-statistic process is introduced and central limittheorems in various norms are obtained for it. This involves the development ofseveral inequalities for $U$-processes that may be useful in other contexts.This local $U$-statistic process is based on an estimator of the density of afunction of several sample variables proposed by Frees [J. Amer. Statist.Assoc. 89 (1994) 517--525] and, as a consequence, uniform in bandwidth centrallimit theorems in the sup and in the $L_p$ norms are obtained for theseestimators.

On local $U$-statistic processes and the estimation of densities of functions of several sample variables