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Methodology and Computing in Applied Probability

, Volume 21, Issue 4, pp 1215–1228 | Cite as

Cutoff Phenomenon for Nearest Lamperti’s Random Walk

  • Wenming Hong
  • Hui YangEmail author
Article
  • 26 Downloads

Abstract

We consider transient neighbor random walks on the positive part of the real line, the transition probability is state dependent being a special case of the Lamperti’s random walk. We show that a sequence of lazy random walks on [0, n] exhibits cutoff phenomenon. As an important step in the proof, we derive the limit speed of the expectation and variance of the hitting times of the random walk exactly. And as a byproduct, we give a probabilistic proof for the law of large numbers of the random walk which has been obtained by Voit (1992) using the method of polynomial hypergroups.

Keywords

Lamperti’s random walk Cutoff Hitting times Law of large numbers Branching structure within the random walk 

Mathematics Subject Classification (2010)

Primary 60J80 Secondary 60G50 

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Notes

Acknowledgments

We thank the anonymous referees for pointing out some useful references and mistakes that helped improve the paper.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical Sciences & Laboratory of Mathematics and Complex SystemsBeijing Normal UniversityBeijingPeople’s Republic of China
  2. 2.College of ScienceMinzu University of ChinaBeijingPeople’s Republic of China

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