On the rate of growth of the overshoot and the maximum partial sum

Philip S Griffin, R. A. Maller

Research output: Contribution to journalArticle

Abstract

Let Tr be the first time at which a random walk Sn escapes from the strip [−r, r], and let |STr| − 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 |STr|, by which we mean that |STr|/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 |STr|/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 |STr|/r → 1 a.s. if and only if EX2 < ∞ and EX = 0 or 0 < |EX| ≤ E|X| < ∞. Proving this requires establishing the equivalence of the stability of STr with certain dominance properties of the maximum partial sum Sn* = max(|Sj| : 1 ≤ j ≤ n) over its maximal increment.

Original languageEnglish (US)
Pages (from-to)181-196
Number of pages16
JournalAdvances in Applied Probability
Volume30
Issue number1
DOIs
StatePublished - 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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