STOCHASTIC PROCESSES AND THEIR APPLICATIONS | 卷:125 |
Limit theorems for random walks that avoid bounded sets, with applications to the largest gap problem | |
Article | |
Vysotsky, Vladislav1,2,3 | |
[1] Arizona State Univ, Tempe, AZ 85287 USA | |
[2] Steklov Math Inst, St Petersburg Dept, Moscow, Russia | |
[3] St Petersburg State Univ, Chebyshev Lab, St Petersburg 199034, Russia | |
关键词: Random walk; Hitting time; Limit theorem; Conditional limit theorem; Harmonic function; Killed random walk; Largest gap; Maximal spacing; Number of non-visited sites; | |
DOI : 10.1016/j.spa.2014.11.017 | |
来源: Elsevier | |
【 摘 要 】
Consider a centred random walk in dimension one with a positive finite variance sigma(2), and let tau(B) be the hitting time for a bounded Borel set B with a non-empty interior. We prove the asymptotic P-x (tau(B) > n) similar to root 2/pi sigma V--1(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, we 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. (C) 2014 Elsevier B.V. All rights reserved.
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