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

TL;DR

This paper investigates the asymptotic behavior of non-homogeneous random walks with heavy-tailed, non-integrable increments, extending classical results to more general, non-Markovian models with applications in risk processes.

Contribution

It generalizes existing results on heavy-tailed random walks by removing the Markov assumption and considering non-homogeneous, non-integrable increments with martingale-based proofs.

Findings

Established transience and bounds for non-homogeneous heavy-tailed walks

Derived conditions for existence of moments of passage times

Applied results to risk models with heavy-tailed increments

Abstract

We study asymptotic properties of spatially non-homogeneous random walks with non-integrable increments, including transience, almost-sure bounds, and existence and non-existence of moments for first-passage and last-exit times. In our proofs we also make use of estimates for hitting probabilities and large deviations bounds. Our results are 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 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.