A Ferguson-Klass-LePage series representation of multistable multifractional processes and related processes

TL;DR

This paper introduces a new series-based construction method for multistable multifractional processes, extending stable process representations to non-stationary, locally controlled processes with practical applications and numerical illustrations.

Contribution

It develops a Ferguson-Klass-LePage series representation for multistable processes, including specific cases like multistable Lévy motion and multifractional motions, with explicit distribution calculations.

Findings

Derived finite-dimensional distributions for multistable processes.

Constructed numerical paths illustrating process behaviors.

Extended stable process representations to non-stationary settings.

Abstract

The study of non-stationary processes whose local form has controlled properties is a fruitful and important area of research, both in theory and applications. We present here a construction of multifractional multistable processes, based on the Ferguson-Klass-LePage series representation of stable processes. We consider various particular cases of interest, including multistable L\'evy motion, multistable reverse Ornstein-Uhlenbeck process, log-fractional multistable motion and linear multistable multifractional motion. We also compute the finite dimensional distributions of those processes. Finally, we display numerical experiments showing graphs of synthesized paths of such processes.