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Cutoff Phenomenon for Nearest Lamperti’s Random Walk

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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.

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Acknowledgments

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

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Authors and Affiliations

  1. School of Mathematical Sciences & Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Wenming Hong

  2. College of Science, Minzu University of China, Beijing, 100081, People’s Republic of China

    Hui Yang

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  1. Wenming Hong
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  2. Hui Yang
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Correspondence to Hui Yang.

Additional information

Supported by NSFC (No.11531001, 11626245) and Fundamental Research Funds for the Central Universities of Minzu University of China (No. 2017QNPY30)

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Hong, W., Yang, H. Cutoff Phenomenon for Nearest Lamperti’s Random Walk. Methodol Comput Appl Probab 21, 1215–1228 (2019). https://doi.org/10.1007/s11009-018-9666-8

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  • DOI: https://doi.org/10.1007/s11009-018-9666-8

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