### Abstract

Let T_{r} be the first time at which a random walk Sn escapes from the strip [−r, r], and let |ST_{r}| − r be the overshoot of the boundary of the strip. We investigate the order of magnitude of the overshoot, as r → ∞, by providing necessary and sufficient conditions for the ‘stability’ of |ST_{r}|, by which we mean that |ST_{r}|/r converges to 1, either in probability (weakly) or almost surely (strongly), as r → ∞. These also turn out to be equivalent to requiring only the boundedness of |ST_{r}|/r, rather than its convergence to 1, either in the weak or strong sense, as r → ∞. The almost sure characterisation turns out to be extremely simple to state and to apply: we have |ST_{r}|/r → 1 a.s. if and only if EX^{2} < ∞ and EX = 0 or 0 < |EX| ≤ E|X| < ∞. Proving this requires establishing the equivalence of the stability of ST_{r} with certain dominance properties of the maximum partial sum S_{n}* = max(|S_{j}| : 1 ≤ j ≤ n) over its maximal increment.

Original language | English (US) |
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Pages (from-to) | 181-196 |

Number of pages | 16 |

Journal | Advances in Applied Probability |

Volume | 30 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1998 |

### Keywords

- Chung law
- Exit time from a strip
- Overshoot
- Random walks
- Renewal theorems
- Stability

### ASJC Scopus subject areas

- Statistics and Probability
- Applied Mathematics

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## Cite this

*Advances in Applied Probability*,

*30*(1), 181-196. https://doi.org/10.1239/aap/1035227999