engleski

Ergodic property of stable-like Markov chains

A stable-like Markov chain is a time-homogeneous Markov chain on the real line with the 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 certain uniformity condition on the density functions $f_x(y)$ and additional mild drift conditions, we give sufficient conditions for recurrence in the case when $0<\liminf_{; ; |x|\longrightarrow\infty}; ; \alpha(x)$, sufficient conditions for transience in the case when $\limsup_{; ; |x|\longrightarrow\infty}; ; \alpha(x) <2$ and sufficient conditions for ergodicity in the case when $0<\inf\ {; ; \alpha(x):x\in\R\}; ; $. 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 $\alpha\neq1.$

ergodicity; Foster-Lyapunov drift criteria; recurrence; stable distribution; stable-like Markov chain; transience

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