### Abstract

Let T_{r} be the first time at which a random walk S_{n} escapes from the strip [-r, r], and let |S_{Tr}| - 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 |S_{Tr}|, by which we mean that \S_{Tr}|/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 |S_{Tr}|/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 |S_{Tr}|/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 S_{Tr} 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 - Mar 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/10.1017/S0001867800008156