## The Annals of Probability

- Ann. Probab.
- Volume 4, Number 2 (1976), 229-248.

### 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 $(\varepsilon > 0)$.

#### Article information

**Source**

Ann. Probab., Volume 4, Number 2 (1976), 229-248.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996131

**Digital Object Identifier**

doi:10.1214/aop/1176996131

**Mathematical Reviews number (MathSciNet)**

MR431388

**Zentralblatt MATH identifier**

0349.60067

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J15

Secondary: 60F15: Strong theorems

**Keywords**

Random walks simple walk multiple range of a walk weak and strong laws of large numbers

#### Citation

Flatto, Leopold. The Multiple Range of Two-Dimensional Recurrent Walk. Ann. Probab. 4 (1976), no. 2, 229--248. doi:10.1214/aop/1176996131. https://projecteuclid.org/euclid.aop/1176996131