Recurrence and transience property for a class of Markov chains

Abstract

We consider the recurrence and transience problem for a time-homogeneous Markov chain on the real line with transition kernel $p(x,\mathrm{d}y)=f_{x}(y-x)\,\mathrm{d}y$, where the density functions $f_{x}(y)$, for large $|y|$, have a power-law decay with exponent $\alpha(x)+1$, where $\alpha(x)\in(0,2)$. In this paper, under a uniformity condition on the density functions $f_{x}(y)$ and an additional mild drift condition, we prove that when $\lim\,\inf_{|x|\longrightarrow\infty}\alpha(x)>1$, the chain is recurrent. Similarly, under the same uniformity condition on the density functions $f_{x}(y)$ and some mild technical conditions, we prove that when $\lim\,\sup_{|x|\longrightarrow\infty}\alpha(x)<1$, the chain is transient. As a special case of these results, we give a new proof for the recurrence and transience property of a symmetric $\alpha$-stable random walk on $\mathbb{R} $ with the index of stability $\alpha\in(0,1)\cup(1,2)$.