Recurrence and transience property for a class of Markov chains

Details Recurrence and transience property for a class of Markov chains Recurrence and transience property for a class of Markov chains

2012-03-02

arxiv.org/abs/1203.0447v1

Mathematics

Probability

Scientific paper

We consider the recurrence and transience problem for a time-homogeneous Markov chain on the real line with 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 uniformity condition on the density functions $f_x(y)$ and some mild technical conditions, we prove that when $\liminf_{|x|\longrightarrow\infty}\alpha(x)>1$, the chain is recurrent, while when $\limsup_{|x|\longrightarrow\infty}\alpha(x)<1$, 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 $\alpha\in(0,1)\cup(1,2).$

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