On the law of the iterated logarithm for canonical U-statistics and processes

The law of the iterated logarithm for canonical or completely degenerate U-statistics with square integrable kernel h is proved, for h taking values in 1, 7 and, in general, in a type 2 separable Banach space. The LIL is also obtained for U-processes indexed by canonical Vapnik-Cervonenkis classes of functions with square integrable envelope and, in this regard, an equicontinuity condition equivalent to the LIL property is quite helpful. Some of these results are then applied to obtain the a.s. exact order of the remainder term in the linearization of the product limit estimator for truncated data; a consequence for density estimation is also included.