Markov chains with heavy-tailed increments and asymptotically zero drift

TL;DR

This paper investigates the recurrence and transience behavior of Markov chains with heavy-tailed increments and near-zero mean, extending classical results to infinite-variance cases across different state spaces.

Contribution

It provides a comprehensive classification of recurrence and transience for Markov chains with heavy-tailed increments and zero drift, including new criteria and moment results.

Findings

Critical regime identified at b3 = b1 - 1 for recurrence/transience

Classification of directional versus oscillatory transience on  space

Sharp conditions for existence of moments of passage times

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 $b + c\pi \textrm{cosec} (\pi \alpha)$. 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.