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Local behaviour of first passage probabilities
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  • Published: 01 December 2010

Local behaviour of first passage probabilities

  • R. A. Doney1 

Probability Theory and Related Fields volume 152, pages 559–588 (2012)Cite this article

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Abstract

Suppose that S is an asymptotically stable random walk with norming sequence c n and that T x is the time that S first enters (x, ∞), where x ≥ 0. The asymptotic behaviour of P(T 0 = n) has been described in a recent paper of Vatutin and Wachtel (Probab. Theory Relat. Fields 143:177–217, 2009), and here we build on that result to give three estimates for P(T x  = n), which hold uniformly as n → ∞ in the regions x = o(c n ), x = O(c n ), and x/c n → ∞, respectively.

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Authors and Affiliations

  1. School of Mathematics, The University of Manchester, Alan Turing Building, Oxford Road, Manchester, M139PL, UK

    R. A. Doney

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  1. R. A. Doney
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Correspondence to R. A. Doney.

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Cite this article

Doney, R.A. Local behaviour of first passage probabilities. Probab. Theory Relat. Fields 152, 559–588 (2012). https://doi.org/10.1007/s00440-010-0330-7

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  • Received: 19 May 2010

  • Revised: 11 October 2010

  • Published: 01 December 2010

  • Issue Date: April 2012

  • DOI: https://doi.org/10.1007/s00440-010-0330-7

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Mathematics Subject Classification (2000)

  • 60G50
  • 60G52
  • 60E07
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