Generalized Hermite processes, discrete chaos and limit theorems

Abstract

We introduce a broad class of self-similar processes {Z(t),t≥0} called generalized Hermite processes. They have stationary increments, are defined on a Wiener chaos with Hurst index H∈(1/2,1), and include Hermite processes as a special case. They are defined through a homogeneous kernel g, called the “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 Zβ(t) of Z(t), which allows H∈(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.