Tempered Hermite process

TL;DR

The paper introduces a tempered Hermite process, which modifies the classical Hermite process with an exponential factor to ensure well-definedness for certain Hurst parameters, and shows it as a limit of discrete chaos processes.

Contribution

It proposes a new tempered Hermite process that extends the Hermite process to broader Hurst parameters and establishes its convergence from discrete chaos processes.

Findings

Defines the tempered Hermite process with exponential tempering.

Proves weak convergence of a discrete chaos process to the tempered Hermite process.

Extends the applicability of Hermite processes to H > 1/2.

Abstract

A tempered Hermite process modifies the power law kernel in the time domain representation of a Hermite process by multiplying an exponential tempering factor $\lambda>0$ such that the process is well defined for Hurst parameter $H>\frac{1}{2}$. A tempered Hermite process is the weak convergence limit of a certain discrete chaos process.