Let $r_n$ be the probability that a recurrent random walk on a countable Abelian group fails to return to the origin in the first $n$ steps. For two-dimensional walk, Kesten and Spitzer have shown that $r_n$ is slowly varying. I.e. $\lim_{n\rightarrow\infty} r_{2n}/r_n = 1$. We strengthen this result and show that for any countable Abelian group of rank 2, $r_n$ is super slowly varying in the sense that $\lim_{n\rightarrow\infty} r_{\lbrack nr_n \rbrack}/r_n = 1$. We use the superslow variation of $r_n$ to obtain the limit law for the number of returns to the origin for all recurrent random walks on these groups.
Publié le : 1975-04-14
Classification:
Random walks,
rank of a countable Abelian group,
super slowly varying sequences,
occupation time theorem,
60J15,
60B15
@article{1176996412,
author = {Flatto, Leopold and Pitt, Joel},
title = {Recurrent Random Walks on Countable Abelian Groups of Rank 2},
journal = {Ann. Probab.},
volume = {3},
number = {6},
year = {1975},
pages = { 380-386},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996412}
}
Flatto, Leopold; Pitt, Joel. Recurrent Random Walks on Countable Abelian Groups of Rank 2. Ann. Probab., Tome 3 (1975) no. 6, pp. 380-386. http://gdmltest.u-ga.fr/item/1176996412/