Laws of the iterated logarithm for the local U-statistic process (Q2385605)

This paper establishes laws of the iterated logarithm for the local U-statistic process. This entails the development of probability inequalities and moment bounds for U-processes that are of independent interest. The local U-statistic process is based upon an estimator of the density of a function of several independent and identically distributed random variables proposed by \textit{E. Frees} [J. Am. Stat. Assoc. 89, No. 426, 517--525 (1994; Zbl 0798.62051)]. As a consequence the results are directly applicable to the derivation of exact rates of uniform in bandwidth consistency in the sup and in the \(L_p\) norms for these estimators.