Limit theorems for random walks that avoid bounded sets, with applications to the largest gap problem

Vladislav Vysotsky

Research output: Contribution to journalArticle

4 Scopus citations

Abstract

Consider a centred random walk in dimension one with a positive finite variance σ2, and let τB be the hitting time for a bounded Borel set B with a non-empty interior. We prove the asymptotic PxB>n)∼2/πσ-1VB(x)n-1/2 and provide an explicit formula for the limit VB as a function of the initial position x of the walk. We also give a functional limit theorem for the walk conditioned to avoid B by the time n. As a main application, we consider the case that B is an interval and study the size of the largest gap Gn (maximal spacing) within the range of the walk by the time n. We prove a limit theorem for Gn, which is shown to be of the constant order, and describe its limit distribution. In addition, we prove an analogous result for the number of non-visited sites within the range of an integer-valued random walk.

Original languageEnglish (US)
Pages (from-to)1886-1910
Number of pages25
JournalStochastic Processes and their Applications
Volume125
Issue number5
DOIs
Publication statusPublished - 2015

    Fingerprint

Keywords

  • Conditional limit theorem
  • Harmonic function
  • Hitting time
  • Killed random walk
  • Largest gap
  • Limit theorem
  • Maximal spacing
  • Number of non-visited sites
  • Random walk

ASJC Scopus subject areas

  • Modeling and Simulation
  • Statistics and Probability
  • Applied Mathematics

Cite this