A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics (Q1062339)

A bounded law of the iterated logarithm for martingales with values in a separable Hilbert space H is proved. It is then applied to prove invariance principles for U-statistics for independent identically distributed (\({\mathbb{R}}\)-valued) random variables \(\{X_ j\), \(j\geq 1\}\) and a kernel \(h: {\mathbb{R}}^ m\to H\), \(m\geq 2\) which is degenerate for the common distribution function of \(X_ j\), \(j\geq 1\). This extends to general m earlier results of the authors, Z. Wahrscheinlichkeitstheor. Verw. Geb. 67, 139-167 (1984; Zbl 0552.60030) on this subject and even gives new results in the case \(H={\mathbb{R}}\).