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We study asymptotic properties of spatially non-homogeneous random walks withnon-integrable increments, including transience, almost-sure bounds, andexistence 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 largedeviations bounds. Our results are more general than existing results in theliterature, which consider only the case of sums of independent (typically,identically distributed) random variables. We do not assume the Markovproperty. Existing results that we generalize include a circle of ideas relatedto the Marcinkiewicz-Zygmund strong law of large numbers, as well as morerecent work of Kesten and Maller. Our proofs are robust and use martingalemethods. We demonstrate the benefit of the generality of our results byapplications to some non-classical models, including random walks withheavy-tailed increments on two-dimensional strips, which include, for instance,certain generalized risk processes.

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