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Passage times of random walks and Lévy processes across power law boundaries
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  • Published: 03 May 2005

Passage times of random walks and Lévy processes across power law boundaries

  • R.A. Doney1 &
  • R.A. Maller2 

Probability Theory and Related Fields volume 133, pages 57–70 (2005)Cite this article

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  • 8 Citations

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Abstract.

We establish an integral test involving only the distribution of the increments of a random walk S which determines whether limsup n →∞(S n /nκ) is almost surely zero, finite or infinite when 1/2<κ<1 and a typical step in the random walk has zero mean. This completes the results of Kesten and Maller [9] concerning finiteness of one-sided passage times over power law boundaries, so that we now have quite explicit criteria for all values of κ≥0. The results, and those of [9], are also extended to Lévy processes.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Manchester, Manchester, M13 9PL, UK

    R.A. Doney

  2. Centre for Mathematical Analysis, and School of Finance and Applied Statistics, Australian National University, Canberra, ACT, Australia

    R.A. Maller

Authors
  1. R.A. Doney
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  2. R.A. Maller
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Corresponding author

Correspondence to R.A. Doney.

Additional information

This work is partially supported by ARC Grant DP0210572.

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Cite this article

Doney, R., Maller, R. Passage times of random walks and Lévy processes across power law boundaries. Probab. Theory Relat. Fields 133, 57–70 (2005). https://doi.org/10.1007/s00440-004-0414-3

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  • Received: 04 May 2004

  • Revised: 17 November 2004

  • Published: 03 May 2005

  • Issue Date: September 2005

  • DOI: https://doi.org/10.1007/s00440-004-0414-3

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Mathematics Subject Classification (2000):

  • Primary 60J30
  • 60K05
  • 60F15
  • 60G40
  • Secondary 60G10
  • 60G17
  • 60J65
  • 60J15

Key words or phrases

  • Random walks
  • Lévy processes
  • Passage times
  • Exit times
  • Ladder processes
  • Power law boundaries
  • Limsup behaviour
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