  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.
