Open Access
2019 Markov chains with heavy-tailed increments and asymptotically zero drift
Nicholas Georgiou, Mikhail V. Menshikov, Dimitri Petritis, Andrew R. Wade
Electron. J. Probab. 24: 1-28 (2019). DOI: 10.1214/19-EJP322


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.


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Nicholas Georgiou. Mikhail V. Menshikov. Dimitri Petritis. Andrew R. Wade. "Markov chains with heavy-tailed increments and asymptotically zero drift." Electron. J. Probab. 24 1 - 28, 2019.


Received: 18 June 2018; Accepted: 18 May 2019; Published: 2019
First available in Project Euclid: 21 June 2019

zbMATH: 07089000
MathSciNet: MR3978212
Digital Object Identifier: 10.1214/19-EJP322

Primary: 60J05
Secondary: 60J10

Keywords: asymptotically zero drift , heavy tails , Lamperti’s problem , Lyapunov functions , passage-time moments , Random walk , recurrence , transience

Vol.24 • 2019
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