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Persistent Random Walks. II. Functional Scaling Limits

Author

Listed:
  • Peggy Cénac

    (Université de Bourgogne Franche-Comté)

  • Arnaud Ny

    (Université Paris Est)

  • Basile Loynes

    (Université de Bretagne-Loire)

  • Yoann Offret

    (Université de Bourgogne Franche-Comté)

Abstract

We describe the scaling limits of the persistent random walks (PRWs) for which the recurrence has been characterized in Cénac et al. (J. Theor. Probab. 31(1):232–243, 2018). We highlight a phase transition phenomenon with respect to the memory: depending on the tails of the persistent time distributions, the limiting process is either Markovian or non-Markovian. In the memoryless situation, the limits are classical strictly stable Lévy processes of infinite variations, but the critical Cauchy case and the asymmetric situation we investigate fill some lacunae of the literature, in particular regarding directionally reinforced random walks (DRRWs). In the non-Markovian case, we extend the results of Magdziarz et al. (Stoch. Process. Appl. 125(11):4021–4038, 2015) on Lévy walks (LWs) to a wider class of PRWs without renewal patterns. Finally, we clarify some misunderstanding regarding the marginal densities in the framework of DRRWs and LWs and compute them explicitly in connection with the occupation times of Lamperti’s stochastic processes.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:2:d:10.1007_s10959-018-0852-y
    DOI: 10.1007/s10959-018-0852-y
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    References listed on IDEAS

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    1. Magdziarz, M. & Scheffler, H.P. & Straka, P. & Zebrowski, P., 2015. "Limit theorems and governing equations for Lévy walks," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4021-4038.
    2. Straka, P. & Henry, B.I., 2011. "Lagging and leading coupled continuous time random walks, renewal times and their joint limits," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 324-336, February.
    3. 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.
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