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April, 1975 Recurrent Random Walks on Countable Abelian Groups of Rank 2
Leopold Flatto, Joel Pitt
Ann. Probab. 3(2): 380-386 (April, 1975). DOI: 10.1214/aop/1176996412


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.


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Leopold Flatto. Joel Pitt. "Recurrent Random Walks on Countable Abelian Groups of Rank 2." Ann. Probab. 3 (2) 380 - 386, April, 1975.


Published: April, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0315.60039
MathSciNet: MR380974
Digital Object Identifier: 10.1214/aop/1176996412

Primary: 60J15
Secondary: 60B15

Keywords: occupation time theorem , Random walks , rank of a countable Abelian group , super slowly varying sequences

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 2 • April, 1975
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