Picture of a sphere with binary code

Making Strathclyde research discoverable to the world...

The Strathprints institutional repository is a digital archive of University of Strathclyde research outputs. It exposes Strathclyde's world leading Open Access research to many of the world's leading resource discovery tools, and from there onto the screens of researchers around the world.

Explore Strathclyde Open Access research content

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

Hryniv, Ostap and MacPhee, I.M. and Menshikov, Mikhail V. and Wade, Andrew (2012) Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips. Electronic Journal of Probability, 17 (59). pp. 1-28. ISSN 1083-6489

[img] PDF (Journal version)
2216_10769_1_PB.pdf - Published Version

Download (340kB)

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.