Self-stabilizing processes based on random signs

K. J. Falconer, J. Lévy Véhel

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

A self-stabilizing processes {Z(t), t ∈ [t0,t1)} is a random process which when localized, that is scaled to a fine limit near a given t ∈ [t0,t1), has the distribution of an α(Z(t))-stable process, where 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 an earlier paper we constructed self-stabilizing processes using sums over plane Poisson point processes in the case of α:ℝ→(0,1) which depended on the almost sure absolute convergence of the sums. Here we construct pure jump self-stabilizing processes when 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 randomizing the point set to get a process with the desired local properties.
Original languageEnglish
Pages (from-to)134-152
Number of pages19
JournalJournal of Theoretical Probability
Volume33
Issue number1
Early online date29 Sept 2018
DOIs
Publication statusPublished - Mar 2020

Keywords

  • Self-similar process
  • Stable process
  • Localisable process
  • Multistable process
  • Poisson point process

Fingerprint

Dive into the research topics of 'Self-stabilizing processes based on random signs'. Together they form a unique fingerprint.

Cite this