Self-stabilizing processes

TL;DR

This paper introduces self-stabilizing processes whose local behavior resembles an alpha-stable process with a stability index depending on the process's current value, constructed via Poisson point processes.

Contribution

It constructs a new class of processes with state-dependent stability indices, extending the theory of stable processes with a novel autoregressive approach.

Findings

The processes exhibit localized alpha-stable behavior with variable stability index.

Construction via Poisson point processes ensures the processes have the desired local distributions.

The approach generalizes existing stable processes to include state-dependent stability.

Abstract

We construct `self-stabilizing' processes {Z(t), t $\in [t_0,t_1)$}. These are random processes which when `localized', that is scaled around t to a fine limit, have the distribution of an $\alpha$(Z(t))-stable process, where $\alpha$ is some given function on R. Thus the stability index at t depends on the value of the process at t. Here we address the case where $\alpha$: R $\to$ (0,1). We first construct deterministic functions which satisfy a kind of autoregressive property involving sums over a plane point set $\Pi$. Taking $\Pi$ to be a Poisson point process then defines a random pure jump process, which we show has the desired localized distributions.