Self-stabilizing processes based on random signs
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.
Citation
Falconer , K J & Lévy Véhel , J 2020 , ' Self-stabilizing processes based on random signs ' , Journal of Theoretical Probability , vol. 33 , no. 1 , pp. 134-152 . https://doi.org/10.1007/s10959-018-0862-9
Publication
Journal of Theoretical Probability
Status
Peer reviewed
ISSN
0894-9840Type
Journal article
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