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 alpha(x)+1, with alpha(x)a(0,2). 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 a(0,2), then the chain is recurrent if and only if alpha+beta a parts per thousand yen2. If the function xa dagger broken vertical bar f (x) is periodic and if the set {x:alpha(x)=alpha (0):=inf (xaa"e) alpha(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 alpha (0)a parts per thousand yen1.