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

2013/12/23 by Vladislav Vysotsky, Vysotsky, Vladislav
Mathematics · #60F17 #60G17 #60G50 #FOS: Mathematics #Functional Equations Stability Results #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60F17 #msc:60G17 #msc:60G50

paper · pdf · doi:10.48550/arxiv.1312.6491

The title changed and some minimal changes added

openalex publication_date 2013/12/23 · arxiv created 2014/01/29 · arxiv updated 2014/01/30 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

Consider a centred random walk in dimension one with a positive finite variance σ², and let τ_B be the hitting time for a bounded Borel set B with a non-empty interior. We prove the asymptotic P_x(τ_B > n) ∼ √(2 / π) σ^-1 V_B(x) n^-1/2 and provide an explicit formula for the limit V_B 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, consider the case that B is an interval and study the size of the largest gap G_n (maximal spacing) within the range of the walk by the time n. We prove a limit theorem for G_n, 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.

Related

• Random walks with square-root boundaries: the case of exact boundaries g(t)=c√(t+b)-a
• Exit times for integrated random walks
• Frequently visited sets for random walks
• How Far Might We Walk at Random?
• Functional limit theorems for random walks