Persistent Random Walks. I. Recurrence Versus Transience

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.

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Acknowledgments

The authors wish to thank the referee for his or her valuable advice and suggestions – especially Remark 3.1 – improving the readability of the exposition and enriching the content of the paper.

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Correspondence to Yoann Offret.

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Cénac, P., Le Ny, A., de Loynes, B. et al. Persistent Random Walks. I. Recurrence Versus Transience. J Theor Probab 31, 232–243 (2018). https://doi.org/10.1007/s10959-016-0714-4

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Keywords

  • Persistent and directionally reinforced random walks
  • Variable length memory
  • Recurrence and transience
  • Random walk with undefined mean

Mathematics Subject Classification (2010)

  • 60G50
  • 60J15
  • 60G17
  • 60J05
  • 37B20
  • 60K35