## Abstract

Renewal-like results and stability theorems relating to the large-time behaviour of a random walk S_{n} reflected in its maximum, R_{n} = max_{0 ≤ j ≤ n} S_{j} - S_{n}, are proved. Mainly, we consider the behaviour of the exit time, τ (r), where τ (r) = min {n ≥ 1 : R_{n} > r}, r > 0, and the exit position, R_{τ (r)}, as r grows large, with particular reference to the cases when S_{n} has finite variance, and/or finite mean. Thus, lim_{r → ∞} E R_{τ (r)} / r = 1 is shown to hold when E | X | < ∞ and E X < 0 or E X^{2} < ∞ and E X = 0, and in these situations E τ (r) grows like a multiple of r, or of r^{2}, respectively. More generally, under only a rather mild side condition, we give equivalences for R_{τ (r)} / r over(→, P) 1 as r → ∞ and lim_{r → ∞} R_{τ (r)} / r = 1 almost surely (a.s.); alternatively expressed, the overshoot R_{τ (r)} - r is o (r) as r → ∞, in probability or a.s. Comparisons are also made with exit times of the random walk S_{n} across both two-sided and one-sided horizontal boundaries.

Original language | English |
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Pages (from-to) | 1270-1297 |

Number of pages | 28 |

Journal | Stochastic Processes and their Applications |

Volume | 119 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 2009 |