An estimation of the stability and the localisability functions of multistable processes

TL;DR

This paper introduces two estimators for the stability and localisability functions of multistable processes, demonstrating their consistency through classical examples like Levy multistable processes and Linear Multifractional Multistable Motion.

Contribution

It provides the first consistent estimators for the stability and localisability functions of multistable processes, advancing analysis of such processes.

Findings

Two consistent estimators are proposed for stability and localisability functions.

Convergence of estimators is demonstrated through classical examples.

The methods are validated on Levy multistable process and Linear Multifractional Multistable Motion.

Abstract

Multistable processes are tangent at each point to a stable process, but where the index of stability and the index of localisability varies along the path. In this work, we give two estimators of the stability and the localisability functions, and we prove the consistency of those two estimators. We illustrate these convergences with two classical examples, the Levy multistable process and the Linear Multifractional Multistable Motion.