How does tempering affect the local and global properties of fractional Brownian motion?

TL;DR

This paper explores how tempering the kernel of fractional Brownian motion influences its local and global properties, revealing that local features remain unchanged while global asymptotic behaviors are significantly affected.

Contribution

It provides a detailed analysis of the effects of tempering on fractional Brownian motion, highlighting the unchanged local properties and altered global asymptotic behaviors.

Findings

Tempering does not alter local properties like sample paths and p-variation.

Tempering significantly impacts the Breuer-Major theorem and asymptotic cumulant behavior.

Global properties are sensitive to tempering, unlike local properties.

Abstract

The present paper investigates the effects of tempering the power law kernel of moving average representation of a fractional Brownian motion (fBm) on some local and global properties of this Gaussian stochastic process. Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) are the processes that are considered in order to investigate the role of tempering. Tempering does not change the local properties of fBm including the sample paths and p-variation, but it has a strong impact on the Breuer-Major theorem, asymptotic behavior of the 3rd and 4th cumulants of fBm and the optimal fourth moment theorem.