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 $$p(x,\hbox {d}y)=f_x(y-x)\hbox {d}y$$ p ( x , d y ) = f x ( y - x ) d y , where the density functions $$f_x(y)$$ f x ( y ) , for large $$|y|$$ | y | , have a power-law decay with exponent $$\alpha (x)+1$$ α ( x ) + 1 , where $$\alpha (x)\in (0,2)$$ α ( x ) ∈ ( 0 , 2 ) . In this paper, under a certain uniformity condition on the density functions $$f_x(y)$$ f x ( y ) and additional mild drift conditions, we give sufficient conditions for recurrence in the case when $$0