Generalized Hermite processes, discrete chaos and limit theorems

TL;DR

This paper introduces generalized Hermite processes with stationary increments and long-range dependence, extending classical Hermite processes through a new kernel, and establishes related limit theorems for both continuous and discrete cases.

Contribution

It defines a new class of self-similar processes called generalized Hermite processes using a generalized kernel, broadening the scope of Hermite processes and their limit theorems.

Findings

Established non-central limit theorems for generalized Hermite processes.

Developed a multivariate limit theorem combining central and non-central limits.

Created a framework for long-range dependent stationary sequences via discrete chaos.

Abstract

We introduce a broad class of self-similar processes $\{Z(t),t\ge 0\}$ called generalized Hermite process. They have stationary increments, are defined on a Wiener chaos with Hurst index $H\in (1/2,1)$, and include Hermite processes as a special case. They are defined through a homogeneous kernel $g$, called "generalized Hermite kernel", which replaces the product of power functions in the definition of Hermite processes. The generalized Hermite kernels $g$ can also be used to generate long-range dependent stationary sequences forming a discrete chaos process $\{X(n)\}$. In addition, we consider a fractionally-filtered version $Z^\beta(t)$ of $Z(t)$, which allows $H\in (0,1/2)$. Corresponding non-central limit theorems are established. We also give a multivariate limit theorem which mixes central and non-central limit theorems.