Recurrence and transience property for a class of Markov chains

TL;DR

This paper investigates the recurrence and transience of a class of Markov chains on the real line with heavy-tailed transition densities, establishing conditions based on tail decay exponents that determine their long-term behavior.

Contribution

It provides new criteria for recurrence and transience of Markov chains with power-law tail transition kernels, extending known results for symmetric stable processes.

Findings

Chains are recurrent if tail decay exponent exceeds 1 at infinity.

Chains are transient if tail decay exponent is less than 1 at infinity.

New proof for recurrence and transience of symmetric alpha-stable random walks.

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 $\alpha\in(0,1)\cup(1,2).$