Multistable processes and localisability

TL;DR

This paper introduces alpha(x)-multistable measures and processes that locally resemble alpha-stable processes with a varying stability index, expanding the modeling capabilities for non-stationary phenomena.

Contribution

It develops a framework for constructing and analyzing multistable processes with a position-dependent stability index using characteristic functions.

Findings

Construction of alpha(x)-multistable measures and processes

Examples demonstrating localisability of these processes

Analysis of how the stability index alpha(x) varies with time

Abstract

We use characteristic functions to construct alpha(x)-multistable measures and integrals, where the measures behave locally like alpha-stable measures, but with the stability index alpha(x) varying with time x. This enables us to construct alpha(x)-multistable processes on R, that is processes whose scaling limit at time x is an alpha(x)-stable process. We present several examples of such multistable processes and examine their localisability.