The LIL for canonical $U$-statistics

Abstract

We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for canonical $U$-statistics of arbitrary order $d$, extending the previously known results for $d=2$. The nasc's are expressed as growth conditions on a parameterized family of norms associated with the $U$-statistics kernel.Comment: Published in at http://dx.doi.org/10.1214/07-AOP351 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org