First-passage times over moving boundaries for asymptotically stable walks

D. Denisov, A. Sakhanenko, V. Wachtel

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


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 languageEnglish
Pages (from-to)613-633
Number of pages21
JournalTheory of Probability and its Applications
Issue number4
Publication statusPublished - 1 Jan 2019


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


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