### Abstract

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 (ℝ-valued) random variables {X_{j}, j≧1} and a kernel h: ℝ^{m}→H, m≧2, which is degenerate for the common distribution function of X_{j}, j≧1. This extends to general m results of an earlier paper on this subject and even gives new results in the case H=ℝ.

Original language | English (US) |
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Pages (from-to) | 111-131 |

Number of pages | 21 |

Journal | Probability Theory and Related Fields |

Volume | 72 |

Issue number | 1 |

DOIs | |

State | Published - Apr 1 1986 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

Dehling, H., Denker, M., & Philipp, W. (1986). A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics.

*Probability Theory and Related Fields*,*72*(1), 111-131. https://doi.org/10.1007/BF00343899