Recurrent Random Walks on Countable Abelian Groups of Rank 2

Abstract

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.