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 $b + c\pi \textrm{ cosec} (\pi \alpha)$. 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.
Categories: math.PR