engleski

Recurrence and transience property of some Markov chains

We consider the recurrence and transience problem for a temporally homogeneous Markov chain on the real line with transition function $p(x, dy)=f_x(y-x)dy$, where the density functions $f_x(y)$, for large $|y|$, have a power-law decay with exponent $\alpha(x)+1$, where $\alpha(x)\in(0, 2)$. Under a uniformity condition on the densities $f_x(y)$ 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. Furthermore, if $f_x(y)$ are densities of symmetric distributions such that the function $x\longmapsto f_x$ is periodic and the set $\ {; ; ; x:\alpha(x)=\alpha_0:=\inf_{; ; ; x\in\R}; ; ; \alpha(x)\}; ; ; $ has positive Lebesgue measure, then, under some mild technical conditions on the densities $f_x(y)$, the chain is recurrent if, and only if, $\alpha_0\geq1.$ Finally, if $f_x(y)$ is the density of a symmetric $\alpha$-stable distribution for negative $x$ and the density of a symmetric $\beta$-stable distribution for non- negative $x$, where $\alpha, \beta\in(0, 2)$, then the chain is recurrent if, and only if, $\alpha+\beta\geq2.$ The same type of results is proved for Markov chains on the integer lattice $\Z$. 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 $\alpha\in(0, 2).$

Foster-Lyapunov drift criterion ; Markov chain ; Markov process ; Feller process ; recurrence ; Harris recurrence ; transience ; characteristics of semamrtingale ; T-model ; stable-like process ; stable distribution

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