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On the asymptotic behaviour of first passage times for transient random walk
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  • Published: June 1989

On the asymptotic behaviour of first passage times for transient random walk

  • R. A. Doney1 

Probability Theory and Related Fields volume 81, pages 239–246 (1989)Cite this article

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Summary

Let τ x denote the time at which a random walk with finite positive mean first passes into (x, ∞), wherex≧0. This paper establishes the asymptotic behaviour of Pr {τ x >n} asn→∞ for fixedx in two cases. In the first case the left hand tail of the step-distribution is regularly varying, and in the second the step-distribution satisfies a one-sided Cramér type condition. As a corollary, it follows that in the first case\(\mathop {\lim }\limits_{n \to \infty } \) Pr {τ x >n}/Pr{τ 0 >n} coincides with the limit of the same quantity for recurrent random walk satisfying Spitzer's condition, but in the second case the limit is more complicated.

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Authors and Affiliations

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

    R. A. Doney

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  1. R. A. Doney
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Doney, R.A. On the asymptotic behaviour of first passage times for transient random walk. Probab. Th. Rel. Fields 81, 239–246 (1989). https://doi.org/10.1007/BF00319553

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  • Received: 03 December 1987

  • Revised: 27 July 1988

  • Issue Date: June 1989

  • DOI: https://doi.org/10.1007/BF00319553

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Keywords

  • Stochastic Process
  • Asymptotic Behaviour
  • Random Walk
  • Probability Theory
  • Mathematical Biology
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