Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips

Ostap Hryniv, Iain M. MacPhee, Mikhail V. Menshikov, and Andrew R. Wade

Electronic Journal of Probability, 17, 2012, paper no. 59. DOI: 10.1214/EJP.v17-2216

Abstract

We study spatially non-homogeneous random walks with non-integrable increments. We consider asymptotic properties including transience, almost-sure bounds, and existence and non-existence of moments for first-passage times and last-exit times. During the course of our proofs we also make use of estimates for hitting probabilities and large deviations bounds. Our results are considerably more general than existing results in the literature, which consider only the case of sums of independent (typically, identically distributed) random variables. We do not even assume the Markov property. Existing results that we generalize include a circle of ideas related to the Marcinkiewicz–Zygmund strong law of large numbers, as well as more recent work of Kesten and Maller. Our proofs are robust and use martingale methods. We demonstrate the benefit of the generality of our results by applications to some non-classical models, including random walks with heavy-tailed increments on two-dimensional strips, which include, for instance, certain generalized risk processes.