Consider an oscillating integer valued random walk up to the first hitting time of some fixed integer x > 0. Suppose there is a fee to be paid each time the random walk crosses the level x, and that the amount corresponds to the overshoot. We determine the distribution of the sum of these fees in terms of the renewal functions of the ascending and descending ladder heights. The proof is based on the observation that some path transformation of the random walk enables us to translate the problem in terms of the intersection of certain regenerative sets.

Bertoin, Jean (1999). *On overshoots and hitting times for random walks.* Journal of Applied Probability, 36(2):593-600.

## Abstract

Consider an oscillating integer valued random walk up to the first hitting time of some fixed integer x > 0. Suppose there is a fee to be paid each time the random walk crosses the level x, and that the amount corresponds to the overshoot. We determine the distribution of the sum of these fees in terms of the renewal functions of the ascending and descending ladder heights. The proof is based on the observation that some path transformation of the random walk enables us to translate the problem in terms of the intersection of certain regenerative sets.

## Citations

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | Random walk; overshoot; ladder height; regenerative set |

Language: | English |

Date: | 1999 |

Deposited On: | 25 Jul 2013 06:44 |

Last Modified: | 05 Apr 2016 16:52 |

Publisher: | Applied Probability Trust |

ISSN: | 0021-9002 |

Publisher DOI: | https://doi.org/10.1239/jap/1032374474 |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=1725429 http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0942.60032 |

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