Markov chains with heavy-tailed increments and asymptotically zero drift
Résumé
We study the recurrence/transience phase transition for Markov chains on R + , R, and R 2 whose increments have heavy tails with exponent in (1, 2) and asymp-totically zero mean. This is the infinite-variance analogue of the classical Lamperti problem. On R + , for example, we show that if the tail of the positive increments is about cy −α for an exponent α ∈ (1, 2) and if the drift at x is about bx −γ , then the critical regime has γ = α − 1 and recurrence/transience is determined by the sign of b + cπ cosec(πα). On R we classify whether transience is directional or oscillatory, and extend an example of Rogozin & Foss to a class of transient mar-tingales which oscillate between ±∞. 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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