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A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics
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  • Published: April 1986

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

  • Herold Dehling1,
  • Manfred Denker1 &
  • Walter Philipp2 

Probability Theory and Related Fields volume 72, pages 111–131 (1986)Cite this article

  • 138 Accesses

  • 21 Citations

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Summary

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: ℝm→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.

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References

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Author information

Authors and Affiliations

  1. Institut für Mathematische Stochastik, Universität Göttingen, Lotzestr. 13, D-3400, Göttingen, Federal Republic of Germany

    Herold Dehling & Manfred Denker

  2. Department of Mathematics, University of Illinois, 1409 West Green Street, 61801, Urbana, IL, USA

    Walter Philipp

Authors
  1. Herold Dehling
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  2. Manfred Denker
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  3. Walter Philipp
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Cite this article

Dehling, H., Denker, M. & Philipp, W. A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics. Probab. Th. Rel. Fields 72, 111–131 (1986). https://doi.org/10.1007/BF00343899

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  • Received: 20 October 1984

  • Revised: 25 September 1985

  • Issue Date: April 1986

  • DOI: https://doi.org/10.1007/BF00343899

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Keywords

  • Distribution Function
  • Hilbert Space
  • Stochastic Process
  • Probability Theory
  • Statistical Theory
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