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

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

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


Authors

Iain M. MacPhee



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.

Citation

Hryniv, O., MacPhee, I. M., Menshikov, M. V., & Wade, A. R. (2012). Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips. Electronic Journal of Probability, 17, Article 59. https://doi.org/10.1214/ejp.v17-2216

Journal Article Type Article
Publication Date Jan 1, 2012
Deposit Date Aug 29, 2012
Publicly Available Date Jan 31, 2013
Journal Electronic Journal of Probability
Publisher Institute of Mathematical Statistics
Peer Reviewed Peer Reviewed
Volume 17
Article Number 59
DOI https://doi.org/10.1214/ejp.v17-2216

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Copyright Statement
This work is licensed under a Creative Commons Attribution 3.0 License.







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