The LIL for canonical $U$-statistics
  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. 
Rados{\l}aw Adamczak, Rafa{\l} Lata{\l}a