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April, 1976 The Multiple Range of Two-Dimensional Recurrent Walk
Leopold Flatto
Ann. Probab. 4(2): 229-248 (April, 1976). DOI: 10.1214/aop/1176996131


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 $(\varepsilon > 0)$.


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Leopold Flatto. "The Multiple Range of Two-Dimensional Recurrent Walk." Ann. Probab. 4 (2) 229 - 248, April, 1976.


Published: April, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0349.60067
MathSciNet: MR431388
Digital Object Identifier: 10.1214/aop/1176996131

Primary: 60J15
Secondary: 60F15

Keywords: multiple range of a walk , Random walks , simple walk , Weak and strong laws of large numbers

Rights: Copyright © 1976 Institute of Mathematical Statistics

Vol.4 • No. 2 • April, 1976
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