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 $(\varepsilon > 0)$.
Publié le : 1976-04-14
Classification:
Random walks,
simple walk,
multiple range of a walk,
weak and strong laws of large numbers,
60J15,
60F15
@article{1176996131,
author = {Flatto, Leopold},
title = {The Multiple Range of Two-Dimensional Recurrent Walk},
journal = {Ann. Probab.},
volume = {4},
number = {6},
year = {1976},
pages = { 229-248},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996131}
}
Flatto, Leopold. The Multiple Range of Two-Dimensional Recurrent Walk. Ann. Probab., Tome 4 (1976) no. 6, pp. 229-248. http://gdmltest.u-ga.fr/item/1176996131/