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2012 Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips
Ostap Hryniv, Iain MacPhee, Mikhail Menshikov, Andrew Wade
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Electron. J. Probab. 17: 1-28 (2012). DOI: 10.1214/EJP.v17-2216

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.

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Ostap Hryniv. Iain MacPhee. Mikhail Menshikov. Andrew Wade. "Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips." Electron. J. Probab. 17 1 - 28, 2012. https://doi.org/10.1214/EJP.v17-2216

Information

Accepted: 2 August 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1261.60070
MathSciNet: MR2959065
Digital Object Identifier: 10.1214/EJP.v17-2216

Subjects:
Primary: 60G07 , 60J05
Secondary: 60F15 , 60G17 , 60G50 , 91B30 (Secondary)

Keywords: Heavy-tailed random walks , last exit times , non-homogeneous random walks , passage times , random walks on strips , random walks with internal degrees of freedom , Rate of escape , Risk process , Semimartingales , transience

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