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Recurrence and transience property for a class of Markov chains

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 an additional mild drift condition, we prove that when $\liminf_{; ; ; ; |x|\longrightarrow\infty}; ; ; ; \alpha(x)>1$, 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 $\limsup_{; ; ; ; |x|\longrightarrow\infty}; ; ; ; \alpha(x)<1$, 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 $\R$ with the index of stability $\alpha\in(0, 1)\cup(1, 2).$

Foster-Lyapunov drift criterion; Harris recurrence; Markov chain; petite set; recurrence; small set; stable distribution; T-chain; transience

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