Self-stabilizing processes based on random signs

TL;DR

This paper introduces a novel construction of self-stabilizing processes with variable stability index, extending previous work to cases where the index exceeds 1, using random signs and point set randomization.

Contribution

It develops a new method for constructing pure jump self-stabilizing processes with variable stability index greater than 1, overcoming convergence issues present in earlier models.

Findings

Successfully constructed processes with variable stability index > 1

Extended previous models to include non-absolutely convergent sums

Demonstrated local stability properties depend on process value at each point

Abstract

A self-stabilizing processes $\{Z(t), t\in [t_0,t_1)\}$ is a random process which when localized, that is scaled to a fine limit near a given $t\in [t_0,t_1)$, has the distribution of an $\alpha(Z(t))$-stable process, where $\alpha: \mathbb{R}\to (0,2)$ 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 $\alpha: \mathbb{R}\to (0,1)$ which depended on the almost sure absolute convergence of the sums. Here we construct pure jump self-stabilizing processes when $\alpha$ 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.