Self-Stabilizing Processes Based on Random Signs

K. J. Falconer () and J. Lévy Véhel ()
Additional contact information
K. J. Falconer: University of St Andrews, North Haugh
J. Lévy Véhel: Université Nantes, Laboratoire de Mathématiques Jean Leray

Journal of Theoretical Probability, 2020, vol. 33, issue 1, 134-152

Abstract: Abstract A self-stabilizing process $$\{Z(t), t\in [t_0,t_1)\}$${Z(t),t∈[t0,t1)} is, a random process which when localised, that is, scaled to a fine limit near a given $$t\in [t_0,t_1)$$t∈[t0,t1), has the distribution of an $$\alpha (Z(t))$$α(Z(t))-stable process, where $$\alpha : {\mathbb {R}}\rightarrow (0,2)$$α:R→(0,2) is a given continuous function. Thus, the stability index near t depends on the value of the process at t. In another paper [5] we constructed self-stabilizing processes using sums over plane Poisson point processes in the case of $$\alpha : {\mathbb {R}}\rightarrow (0,1)$$α:R→(0,1) which depended on the almost sure absolute convergence of the sums. Here we construct pure jump self-stabilizing processes when $$\alpha $$α may take values greater than 1 when convergence may no longer be absolute. We do this in two stages, firstly by setting up a process based on a fixed point set but taking random signs of the summands, and then randomising the point set to get a process with the desired local properties.

Keywords: Self-similar process; Stable process; Localisable process; Multistable process; Poisson point process; 60G18; 60G52 (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-018-0862-9 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:33:y:2020:i:1:d:10.1007_s10959-018-0862-9

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

DOI: 10.1007/s10959-018-0862-9

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 ().