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 Lp norms are obtained for these estimators.
"On local U-statistic processes and the estimation of densities of functions of several sample variables." Ann. Statist. 35 (3) 1105 - 1145, July 2007. https://doi.org/10.1214/009053607000000154