Recurrence and transience property for a class of Markov chains

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 $\lim {\inf }_{|x|⟶\infty }\alpha (x)>1$, 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 $\lim {\sup }_{|x|⟶\infty }\alpha (x)<1$, 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 $\alpha \in (0,1)\cup (1,2).$

arXiv commentPublished in at http://dx.doi.org/10.3150/12-BEJ448 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (1)

• An Overshoot Approach to Recurrence and Transience of Markov Processes

Cited by in corpus (2)

• Heavy-tailed random walks on complexes of half-lines
• Markov chains with heavy-tailed increments and asymptotically zero drift