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Persistent Random Walks. I. Recurrence Versus Transience

Author

Listed:
  • Peggy Cénac

    (Université de Bourgogne-Franche-Conté (UBFC))

  • Arnaud Ny

    (Université Paris-Est Créteil (UPEC))

  • Basile Loynes

    (Université de Strasbourg)

  • Yoann Offret

    (Université de Bourgogne-Franche-Conté (UBFC))

Abstract

We consider a walker on the line that at each step keeps the same direction with a probability which depends on the time already spent in the direction the walker is currently moving. These walks with memories of variable length can be seen as generalizations of directionally reinforced random walks introduced in Mauldin et al. (Adv Math 117(2):239–252, 1996). We give a complete and usable characterization of the recurrence or transience in terms of the probabilities to switch the direction and we formulate some laws of large numbers. The most fruitful situation emerges when the running times both have an infinite mean. In that case, these properties are related to the behaviour of some embedded random walk with an undefined drift so that these features depend on the asymptotics of the distribution tails related to the persistence times. In the other case, the criterion reduces to a null-drift condition. Finally, we deduce some criteria for a wider class of persistent random walks whose increments are encoded by a variable length Markov chain having—in full generality—no renewal pattern in such a way that their study does not reduce to a skeleton RW as for the original model.

Suggested Citation

  • Peggy Cénac & Arnaud Ny & Basile Loynes & Yoann Offret, 2018. "Persistent Random Walks. I. Recurrence Versus Transience," Journal of Theoretical Probability, Springer, vol. 31(1), pages 232-243, March.
    • Handle: RePEc:spr:jotpro:v:31:y:2018:i:1:d:10.1007_s10959-016-0714-4
    DOI: 10.1007/s10959-016-0714-4
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    References listed on IDEAS

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    1. Weiss, George H, 2002. "Some applications of persistent random walks and the telegrapher's equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 311(3), pages 381-410.
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    Cited by:

    1. Peggy Cénac & Arnaud Ny & Basile Loynes & Yoann Offret, 2019. "Persistent Random Walks. II. Functional Scaling Limits," Journal of Theoretical Probability, Springer, vol. 32(2), pages 633-658, June.

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