On operator fractional L\'evy motion: integral representations and time reversibility

TL;DR

This paper introduces operator fractional Lévy motion (ofLm), a versatile class of non-Gaussian stochastic processes with self-similarity and stationary increments, extending previous models and analyzing their integral representations and time reversibility.

Contribution

It constructs the ofLm class, provides integral representations, and characterizes conditions for time reversibility, especially highlighting differences from Gaussian cases.

Findings

ofLm generalizes univariate and multivariate fractional Brownian motion

Integral representations are established in time and Fourier domains

Time reversibility conditions depend on the Lévy measure and are more restrictive in non-Gaussian cases

Abstract

In this paper, we construct operator fractional L\'evy motion (ofLm), a broad class of non-Gaussian stochastic processes that are covariance operator self-similar, have wide-sense stationary increments and display infinitely divisible marginal distributions. The ofLm class generalizes the univariate fractional L\'evy motion as well as the multivariate operator fractional Brownian motion (ofBm). The ofLm class can be divided into two types, namely, moving average (maofLm) and real harmonizable (rhofLm), both of which share the covariance structure of ofBm under assumptions. We show that maofLm and rhofLm admit stochastic integral representations in the time and Fourier domains, and establish their distinct small- and large-scale limiting behavior. We characterize time reversibility for ofLm through parametric conditions related to its L\'evy measure, starting from a framework for the uniqueness of finite second moment, multivariate stochastic integral representations. In particular, we show that, under non-Gaussianity, the parametric conditions for time reversibility are generally more restrictive than those for the Gaussian case (ofBm).