Markov chains with heavytailed 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 infinitevariance 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/nonexistence of moments of passage times.
 Publication:

arXiv eprints
 Pub Date:
 June 2018
 arXiv:
 arXiv:1806.07166
 Bibcode:
 2018arXiv180607166G
 Keywords:

 Mathematics  Probability
 EPrint:
 Electronic Journal of Probability, Vol. 24 (2019), article 62