### Abstract

Let {S
_{n}
,n≥ 1} be a random walk with independent and identically distributed increments, and let {g
_{n}
,n≥ 1} be a sequence of real numbers. Let T
_{g}
denote the first time when S
_{n}
leaves (g
_{n}
, ∞). Assume that the random walk is oscillating and asymptotically stable, that is, there exists a sequence {c
_{n}
,n≥ 1} such that S
_{n}
/c
_{n}
converges to a stable law. In this paper we determine the tail behavior of T
_{g}
for all oscillating asymptotically stable walks and all boundary sequences satisfying g
_{n}
= o(c
_{n}
). Furthermore, we prove that the rescaled random walk conditioned to stay above the boundary up to time n converges, as n →∞, towards the stable meander.

Original language | English |
---|---|

Pages (from-to) | 613-633 |

Number of pages | 21 |

Journal | Theory of Probability and its Applications |

Volume | 63 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

### Keywords

- First-passage time
- Moving boundary
- Overshoot
- Random walk
- Stable distribution
- random walk
- overshoot
- first-passage time
- stable distribution
- moving boundary

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## Cite this

*Theory of Probability and its Applications*,

*63*(4), 613-633. https://doi.org/10.1137/S0040585X97T989283