Ergodic Property of Stable-Like Markov Chains

Abstract

A stable-like Markov chain is a time-homogeneous Markov chain on the real line with the transition kernel $p(x,\hbox {d}y)=f_x(y-x)\hbox {d}y$, 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 certain uniformity condition on the density functions $f_x(y)$ and additional mild drift conditions, we give sufficient conditions for recurrence in the case when $0<\liminf _{|x|\longrightarrow \infty }\alpha (x)$, sufficient conditions for transience in the case when $\limsup _{|x|\longrightarrow \infty }\alpha (x)<2$ and sufficient conditions for ergodicity in the case when $0<\inf \{\alpha (x):x\in \mathbb {R}\}$. 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 $\alpha \ne 1$.