Markov chains with heavy-tailed increments and asymptotically zero drift

Abstract

We study the recurrence/transience phase transition for Markov chains on ${\mathbb{R} }_{+}$, $\mathbb{R} $, and ${\mathbb{R} }^{2}$ 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 ${\mathbb{R} }_{+}$, for example, we show that if the tail of the positive increments is about $c y^{-\alpha }$ for an exponent $\alpha \in (1,2)$ and if the drift at $x$ is about $b x^{-\gamma }$, then the critical regime has $\gamma = \alpha -1$ and recurrence/transience is determined by the sign of $b + c\pi \operatorname{cosec} (\pi \alpha )$. On $\mathbb{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 $\pm \infty $. In addition to our recurrence/transience results, we also give sharp results on the existence/non-existence of moments of passage times.