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Journal of Theoretical Probability

, Volume 31, Issue 3, pp 1900–1922 | Cite as

Conservative and Semiconservative Random Walks: Recurrence and Transience

  • Vyacheslav M. Abramov
Article

Abstract

In the present paper, we define conservative and semiconservative random walks in \(\mathbb {Z}^d\) and study various families of random walks. The family of symmetric random walks is one of the families of conservative random walks, and simple (Pólya) random walks are their representatives. The classification of random walks given in the present paper enables us to provide a new approach to random walks in \(\mathbb {Z}^d\) by reduction to birth-and-death processes. We construct nontrivial examples of recurrent random walks in \(\mathbb {Z}^d\) for any \(d\ge 3\) and transient random walks in \(\mathbb {Z}^2\).

Keywords

Multi-dimensional random walks Birth-and-death process Markovian single-server queues 

Mathematics Subject Classification (2010)

60G50 60J80 60C05 60K25 

Notes

Acknowledgements

The author thanks the anonymous referee for valuable comments leading to substantial improvement of the paper.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesMonash UniversityClaytonAustralia
  2. 2.School of ScienceRoyal Melbourne Institute of TechnologyMelbourneAustralia

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