Positivity of integrated random walks

Vladislav Vysotsky

Research output: Contribution to journalArticle

6 Scopus citations

Abstract

Take a centered random walk Sn and consider the sequence of its partial sums An := ∑ni=1 Si. Suppose S1 is in the domain of normal attraction of an α-stable law with 1 < α ≤ 2. Assuming that S1 is either right-exponential (i.e. P(S1 < x|S1 > 0) = e -ax for some a > 0 and all x > 0) or right-continuous (skip free), we prove that P{A1 > 0, . . . , AN > 0} ~ CαN1/(2α)-1/2 as N → ∞, where Cα > 0 depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

Original languageEnglish (US)
Pages (from-to)195-213
Number of pages19
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume50
Issue number1
DOIs
StatePublished - Feb 2014

Keywords

  • Area of excursion
  • Area of random walk
  • Integrated random walk
  • One-sided exit probability
  • Persistence
  • Sparre-Andersen theorem
  • Stable excursion
  • Unilateral small deviations

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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