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

A notion of local $U$-statistic process is introduced and central limit theorems in various norms are obtained for it. This involves the development of several 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 a function of several sample variables proposed by Frees [J. Amer. Statist. Assoc. 89 (1994) 517–525] and, as a consequence, uniform in bandwidth central limit theorems in the sup and in the $L_p$ norms are obtained for these estimators.