On the probability that integrated random walks stay positive

Vladislav Vysotsky

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


Let Sn be a centered random walk with a finite variance, and consider the sequence An:=∑i=1n S i, which we call an integrated random walk. We are interested in the asymptotics of PN:=P{min1≤k≤N Ak≥0 as N→∞. Sinai (1992) [15] proved that pN equivalent to N-1/4 if Sn is a simple random walk. We show that pN equivalent to N -1/4 for some other kinds of random walks that include double-sided exponential and double-sided geometric walks, both not necessarily symmetric. We also prove that pN ≤ cN-1/4 for integer-valued walks and upper exponential walks, which are the walks such that Law(S1|S 1>0) is an exponential distribution.

Original languageEnglish (US)
Pages (from-to)1178-1193
Number of pages16
JournalStochastic Processes and their Applications
Issue number7
StatePublished - Jul 2010
Externally publishedYes


  • Area of excursion
  • Area of random walk
  • Excursion
  • Integrated random walk
  • One-sided exit probability
  • Unilateral small deviations

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics


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