Persistent Random Walks. II. Functional Scaling Limits

Peggy Cénac (), Arnaud Ny (), Basile Loynes () and Yoann Offret ()
Additional contact information
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

Journal of Theoretical Probability, 2019, vol. 32, issue 2, 633-658

Abstract: 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.

Keywords: Persistent random walks; Functional scaling limits; Arcsine Lamperti laws; Directionally reinforced random walks; Lévy walks; Anomalous diffusions; 60F17; 60G50; 60J15; 60G17; 60J05; 60G22; 60K20 (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-018-0852-y Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:32:y:2019:i:2:d:10.1007_s10959-018-0852-y

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-018-0852-y

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().