Journal of Theoretical Probability

, Volume 32, Issue 2, pp 633–658 | Cite as

Persistent Random Walks. II. Functional Scaling Limits

  • Peggy Cénac
  • Arnaud Le Ny
  • Basile de Loynes
  • Yoann OffretEmail author


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.


Persistent random walks Functional scaling limits Arcsine Lamperti laws Directionally reinforced random walks Lévy walks Anomalous diffusions 

Mathematics Subject Classification (2010)

60F17 60G50 60J15 60G17 60J05 60G22 60K20 

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Authors and Affiliations

  1. 1.Institut de Mathématiques de Bourgogne (IMB) - UMR CNRS 5584Université de Bourgogne Franche-ComtéDijonFrance
  2. 2.EnsaiUniversité de Bretagne-LoireBRUZ CedexFrance
  3. 3.Laboratoire d’Analyse et de Mathématiques Appliquées (LAMA) - UMR CNRS 8050Université Paris EstCréteil CedexFrance

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