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Recurrence and transience property for a class of Markov chains. (English) Zbl 1284.60090

Summary: 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 \(\lim\,\inf_{|x|\longrightarrow\infty}\alpha(x)>1\), 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 \(\lim\,\sup_{|x|\longrightarrow\infty}\alpha(x)<1\), 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 \(\alpha\in(0,1)\cup(1,2\)).

MSC:

60G50 Sums of independent random variables; random walks
60J05 Discrete-time Markov processes on general state spaces

References:

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