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Local limit theorem for the maximum of asymptotically stable random walks
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  • Published: 09 October 2010

Local limit theorem for the maximum of asymptotically stable random walks

  • Vitali Wachtel1 

Probability Theory and Related Fields volume 152, pages 407–424 (2012)Cite this article

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  • 7 Citations

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Abstract

Let {S n ; n ≥ 0} be an asymptotically stable random walk and let M n denote it’s maximum in the first n steps. We show that the asymptotic behaviour of local probabilities for M n can be approximated by the density of the maximum of the corresponding stable process if and only if the renewal mass-function based on ascending ladder heights is regularly varying at infinity. We also give some conditions on the random walk, which guarantee the desired regularity of the renewal mass-function. Finally, we give an example of a random walk, for which the local limit theorem for M n does not hold.

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Authors and Affiliations

  1. Mathematical Institute, University of Munich, Theresienstrasse 39, 80333, Munich, Germany

    Vitali Wachtel

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  1. Vitali Wachtel
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Correspondence to Vitali Wachtel.

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Wachtel, V. Local limit theorem for the maximum of asymptotically stable random walks. Probab. Theory Relat. Fields 152, 407–424 (2012). https://doi.org/10.1007/s00440-010-0326-3

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  • Received: 04 May 2010

  • Revised: 09 August 2010

  • Published: 09 October 2010

  • Issue Date: April 2012

  • DOI: https://doi.org/10.1007/s00440-010-0326-3

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Keywords

  • Limit theorems
  • Random walks
  • Renewal theorem

Mathematics Subject Classification (2000)

  • 60G50
  • 60G52
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