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Hitting time of a half-line by two-dimensional random walk
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  • Published: 02 January 2004

Hitting time of a half-line by two-dimensional random walk

  • Yasunari Fukai1 

Probability Theory and Related Fields volume 128, pages 323–346 (2004)Cite this article

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  • 7 Citations

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Abstract.

We consider the probability that a two-dimensional random walk starting from the origin never returns to the half-line {(x 1 ,x 2 )|x 1 ≤0,x 2 =0} before time n. It is proved that for aperiodic random walk with mean zero and finite 2+δ(>2)-th absolute moment, this probability times n 1/4 converges to some positive constant c * as \({{n \rightarrow \infty}}\). We show that c * is expressed by using the characteristic function of the increment of the random walk. For the simple random walk, this expression gives \({{ c^{{*}} = \sqrt{{1+ \sqrt{{2}}}}/(2 \Gamma (3/4)).}}\)

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References

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Authors and Affiliations

  1. Fukuokaculty of Mathematics, Kyushu University, 6-10-1, Hakozaki, Higashi-Ku, Fukuoka, 812-8581, Japan

    Yasunari Fukai

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  1. Yasunari Fukai
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Correspondence to Yasunari Fukai.

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Mathematics Subject Classification (2000): 60G50, 60E10

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Cite this article

Fukai, Y. Hitting time of a half-line by two-dimensional random walk. Probab. Theory Relat. Fields 128, 323–346 (2004). https://doi.org/10.1007/s00440-003-0306-y

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  • Received: 06 April 2003

  • Revised: 18 September 2003

  • Published: 02 January 2004

  • Issue Date: March 2004

  • DOI: https://doi.org/10.1007/s00440-003-0306-y

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Keywords

  • Two-dimensional random walk
  • Hitting probability
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