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

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  • Vysotsky, Vladislav

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 Px(τB>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.

Suggested Citation

  • Vysotsky, Vladislav, 2015. "Limit theorems for random walks that avoid bounded sets, with applications to the largest gap problem," Stochastic Processes and their Applications, Elsevier, vol. 125(5), pages 1886-1910.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:5:p:1886-1910
    DOI: 10.1016/j.spa.2014.11.017
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    Cited by:

    1. Uchiyama, Kôhei, 2019. "Asymptotically stable random walks of index 1<α<2 killed on a finite set," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5151-5199.
    2. Döring, Leif & Kyprianou, Andreas E. & Weissmann, Philip, 2020. "Stable processes conditioned to avoid an interval," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 471-487.
    3. Micha Buck, 2021. "Limit Theorems for Random Walks with Absorption," Journal of Theoretical Probability, Springer, vol. 34(1), pages 241-263, March.

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