Ergodic property of stable-like Markov chains

TL;DR

This paper investigates the ergodic, recurrent, and transient behaviors of stable-like Markov chains on the real line, providing new sufficient conditions based on the tail decay exponents and mild drift assumptions.

Contribution

It establishes novel sufficient conditions for recurrence, transience, and ergodicity of stable-like Markov chains with variable tail decay exponents, extending classical results.

Findings

Provided criteria for recurrence when tail decay exponent's lim inf is positive.

Established conditions for transience when the lim sup of the tail decay exponent is less than 2.

Proved ergodicity under a uniform lower bound on the tail decay exponent.

Abstract

A stable-like Markov chain is a time-homogeneous Markov chain on the real line with the transition kernel $p(x,dy)=f_x(y-x)dy$, 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 $\alpha\neq1.$