Recurrence and transience property for a class of Markov chains

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 $α(x)+1$, where $α(x)\in(0,2)$. In this paper, under a uniformity condition on the density functions $f_x(y)$ and an additional mild drift condition, we prove that when $\lim\inf_{|x|\longrightarrow\infty}α(x)>1$, 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 $\lim\sup_{|x|\longrightarrow\infty}α(x)<1$, 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 $α$-stable random walk on $\mathbb {R}$ with the index of stability $α\in(0,1)\cup(1,2).$

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