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An almost sure functional limit theorem at zero for a class of Lévy processes normed by the square root function, and applications
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  • Published: 16 October 2007

An almost sure functional limit theorem at zero for a class of Lévy processes normed by the square root function, and applications

  • Boris Buchmann1,
  • Ross Maller2 &
  • Alex Szimayer3 

Probability Theory and Related Fields volume 142, pages 219–247 (2008)Cite this article

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  • 5 Citations

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Abstract

A recent result of Bertoin, Doney and Maller (Ann. Prob., 2007) gives an integral condition to characterize the class of Lévy processes X(t) for which lim sup\(_{t\downarrow 0}|X(t)|/\sqrt{t} \in (0,\infty)\) occurs almost surely (a.s.). For such processes we have a kind of almost sure “iterated logarithm” result, but without the logs. In the present paper we prove a functional version of this result, which then opens the way to various interesting applications obtained via a continuous mapping theorem. We set these out in a rigorous framework, including a characterisation of the existence of an a.s. cluster set for the interpolated process, appropriate to the continuous time situation. The applications relate to functional laws for the supremum, reflected and a variety of other processes, including a class of stochastic differential equations, where we aim to give as informative a description as we can of the functional limit sets.

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Author information

Authors and Affiliations

  1. Room 348, Building 28, School of Mathematics, Monash University, Clayton Campus, Clayton, VIC, 3800, Australia

    Boris Buchmann

  2. Centre for Mathematics and its Applications, School of Finance and Applied Statistics, Australian National University, Canberra, ACT, 0200, Australia

    Ross Maller

  3. Department of Financial Mathematics, Fraunhofer ITWM, Fraunhofer-Platz 1, 67663, Kaiserslautern, Germany

    Alex Szimayer

Authors
  1. Boris Buchmann
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  2. Ross Maller
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  3. Alex Szimayer
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Corresponding author

Correspondence to Boris Buchmann.

Additional information

This research was partially supported by ARC grant DP0664603.

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Cite this article

Buchmann, B., Maller, R. & Szimayer, A. An almost sure functional limit theorem at zero for a class of Lévy processes normed by the square root function, and applications. Probab. Theory Relat. Fields 142, 219–247 (2008). https://doi.org/10.1007/s00440-007-0103-0

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  • Received: 29 August 2006

  • Revised: 16 August 2007

  • Published: 16 October 2007

  • Issue Date: September 2008

  • DOI: https://doi.org/10.1007/s00440-007-0103-0

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Keywords

  • Lévy process
  • Local behaviour
  • Almost sure convergence
  • Strassen’s functional LIL
  • Iterated logarithm laws

Mathematics Subject Classification (2000)

  • 60G51
  • 60F15
  • 60F17
  • 60F05
  • 60J65
  • 60J75
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