Recurrence and transience criteria for two cases of stable-like Markov chains

TL;DR

This paper establishes criteria for recurrence and transience of certain stable-like Markov chains on the real line, based on the decay exponents of their transition densities, with implications for understanding their long-term behavior.

Contribution

It provides new recurrence and transience criteria for stable-like Markov chains with position-dependent stable distribution densities, extending previous results to more general cases.

Findings

Chain is recurrent if and only if ppa + eta  2 for stable distributions

Periodic case: chain is recurrent if and only if ppa_0  1, where ppa_0 is the minimal stability index

Criteria depend on the decay exponents of the transition densities and their distributional properties.

Abstract

We give recurrence and transience criteria for two cases of time-homogeneous Markov chains on the real line with transition kernel $p(x,dy)=f_x(y-x)dy$, where $f_x(y)$ are probability densities of symmetric distributions and, for large $|y|$, have a power-law decay with exponent $\alpha(x)+1$, with $\alpha(x)\in(0,2)$. If $f_x(y)$ is the density of a symmetric $\alpha$-stable distribution for negative $x$ and the density of a symmetric $\beta$-stable distribution for non-negative $x$, where $\alpha,\beta\in(0,2)$, then the chain is recurrent if and only if $\alpha+\beta\geq2.$ If the function $x\longmapsto f_x$ is periodic and if the set $\{x:\alpha(x)=\alpha_0:=\inf_{x\in\R}\alpha(x)\}$ has positive Lebesgue measure, then, under a uniformity condition on the densities $f_x(y)$ and some mild technical conditions, the chain is recurrent if and only if $\alpha_0\geq1.$