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

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.