Giné, Evarist et M. Mason, David, On Local U-Statistic Processes and the Estimation of Densities of Functions of Several Sample Variables, Annals of statistics , 35(3), 2007, pp. 1105-1145
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.