The Multiple Range of Two-Dimensional Recurrent Walk

Abstract

For each positive integer $p$, let $R_n^p$ be the number of points visited exactly $p$ times by a random walk during the course of its first $n$ steps. We call the random variables $R_n^p$ the multiple range of order $p$ for the given walk. We prove that for two-dimensional simple walk, $R_n^p$ obeys the strong law of large numbers $\lim_{n\rightarrow\infty} R_n^p/(\pi^2 n/\log^2 n) = 1\mathrm{a.s.}$ The method of proof generalizes to yield a similar result for all genuine two-dimensional walks with 0 mean and finite $2 + \varepsilon$ moments $(\varepsilon > 0)$.