### 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) |
---|---|

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 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

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

### Cite this

*Probability Theory and Related Fields*,

*72*(1), 111-131. https://doi.org/10.1007/BF00343899

}

*Probability Theory and Related Fields*, vol. 72, no. 1, pp. 111-131. https://doi.org/10.1007/BF00343899

**A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics.** / Dehling, Herold; Denker, Manfred; Philipp, Walter.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics

AU - Dehling, Herold

AU - Denker, Manfred

AU - Philipp, Walter

PY - 1986/4/1

Y1 - 1986/4/1

N2 - 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 {Xj, j≧1} and a kernel h: ℝm→H, m≧2, which is degenerate for the common distribution function of Xj, 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.

AB - 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 {Xj, j≧1} and a kernel h: ℝm→H, m≧2, which is degenerate for the common distribution function of Xj, 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.

UR - http://www.scopus.com/inward/record.url?scp=0000986469&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000986469&partnerID=8YFLogxK

U2 - 10.1007/BF00343899

DO - 10.1007/BF00343899

M3 - Article

AN - SCOPUS:0000986469

VL - 72

SP - 111

EP - 131

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 1

ER -