Recurrence and Transience Criteria for Two Cases of Stable-Like Markov Chains

We give recurrence and transience criteria for two cases of time-homogeneous Markov chains on the real line with transition kernel p(x,dy)=f x (y−x) dy, where f x (y) are probability densities of symmetric distributions and, for large |y|, have a power-law decay with exponent α(x)+1, with α(x)∈(0,2). If f x (y) is the density of a symmetric α-stable distribution for negative x and the density of a symmetric β-stable distribution for non-negative x, where α,β∈(0,2), then the chain is recurrent if and only if α+β≥2. If the function x↦f x is periodic and if the set {x:α(x)=α 0:=inf x∈ℝ α(x)} has positive Lebesgue measure, then, under a uniformity condition on the densities f x (y) and some mild technical conditions, the chain is recurrent if and only if α 0≥1.