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.
"The LIL for canonical U-statistics." Ann. Probab. 36 (3) 1023 - 1058, May 2008. https://doi.org/10.1214/07-AOP351