The LIL for canonical \(U\)-statistics (Q2427056)

This paper gives necessary and sufficient conditions for the (bounded) law of the iterated logarithm for canonical \(U\)-statistics of arbitrary order \(d\). This extends the previous results developed for the case \(d=2\) in [\textit{E. Giné, S. Kwapien, R. Latała} and \textit{J. Zinn} [Ann. Probab. 29, 520--557 (2001; Zbl 1014.60026)]. The conditions are expressed as growth conditions on a parameterized family of norms associated with the \(U\)-statistic's kernel. The result also proves that a conjecture in the above mentioned paper is false.