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

作者：Vysotsky Vladislav^*

来源：Stochastic Processes and Their Applications, 2015, 125(5): 1886-1910.

DOI：10.1016/j.spa.2014.11.017

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

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.

• 出版日期2015-5