Markov chains with heavy-tailed increments and asymptotically zero drift

We study the recurrence/transience phase transition for Markov chains on R+, R, and R-2 whose increments have heavy tails with exponent in (1, 2) and asymptotically zero mean. This is the infinite-variance analogue of the classical Lamperti problem. On R+, for example, we show that if the tail of the positive increments is about cy(-alpha) for an exponent alpha is an element of (1, 2) and if the drift at x is about bx(-gamma), then the critical regime has gamma = alpha - 1 and recurrence/transience is determined by the sign of b + c pi cosec(pi alpha). On R we classify whether transience is directional or oscillatory, and extend an example of Rogozin & Foss to a class of transient martingales which oscillate between +/-infinity. In addition to our recurrence/transience results, we also give sharp results on the existence/non-existence of moments of passage times.